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2018 AIME I Problems/Problem 10 - Revision history
2024-03-29T02:28:05Z
Revision history for this page on the wiki
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Sails: I made a new solution since when mocking this problem I felt that using recursion would be too bashy. I'm sorry if there are any errors :D
2024-03-28T22:00:37Z
<p>I made a new solution since when mocking this problem I felt that using recursion would be too bashy. I'm sorry if there are any errors :D</p>
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<td colspan="2" style="background-color: #fff; color: #222; text-align: center;">Revision as of 22:00, 28 March 2024</td>
</tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l133" >Line 133:</td>
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<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>~[https://artofproblemsolving.com/wiki/index.php/User:Magnetoninja Magnetoninja]</div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>~[https://artofproblemsolving.com/wiki/index.php/User:Magnetoninja Magnetoninja]</div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">==Solution 6(Generating Functions???)==</ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">We can view the diagram as a collection of points on the complex plane that are connected by transformations/symmetries. The inner circle can be represented as the fifth roots of unity(WLOG let <math>A</math> correspond to <math>1</math>) where moving between the inner circle can be mapped to multiplying by <math>e^{\frac{2i\pi}{5}}</math> we can also view moving between circles as a multiplying by <math>-e^{-2i\pi} = -1.</math> </ins></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">Therefore the generating function for the paths beginning at <math>1</math> or <math>A</math> can be seen as <math>(e^{\frac{2i\pi}{5}} - 1)^{15}.</math></ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;"></ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">We want the positive real terms or the ones that end on <math>1</math> and those add up to <math>\dbinom{15}{0} + \dbinom{15}{10} = 3004 \equiv \boxed{004} \pmod{1000}.</math></ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;"></ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">~SailS</ins></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>==See Also==</div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>==See Also==</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>{{AIME box|year=2018|n=I|num-b=9|num-a=11}}</div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>{{AIME box|year=2018|n=I|num-b=9|num-a=11}}</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>{{MAA Notice}}</div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>{{MAA Notice}}</div></td></tr>
</table>
Sails
https://artofproblemsolving.com/wiki/index.php?title=2018_AIME_I_Problems/Problem_10&diff=206753&oldid=prev
Potato2017: /* Solution 5 (Stars and Bars) */ - typo
2023-12-10T04:37:17Z
<p><span dir="auto"><span class="autocomment">Solution 5 (Stars and Bars): </span> - typo</span></p>
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<td colspan="2" style="background-color: #fff; color: #222; text-align: center;">Revision as of 04:37, 10 December 2023</td>
</tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l118" >Line 118:</td>
<td colspan="2" class="diff-lineno">Line 118:</td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>Case 4: 3 pairs</div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>Case 4: 3 pairs</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>Again, the only solutions for <math>(C,C')</math> are <math>(2,7)</math> and <math>(7,2)</math>. Using stars and bars, we obtain <math>360+<del class="diffchange diffchange-inline">730</del>=1080</math>.</div></td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>Again, the only solutions for <math>(C,C')</math> are <math>(2,7)</math> and <math>(7,2)</math>. Using stars and bars, we obtain <math>360+<ins class="diffchange diffchange-inline">720</ins>=1080</math>.</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>Case 5: 4 pairs</div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>Case 5: 4 pairs</div></td></tr>
</table>
Potato2017
https://artofproblemsolving.com/wiki/index.php?title=2018_AIME_I_Problems/Problem_10&diff=194310&oldid=prev
Magnetoninja: /* Solution 5 (Stars and Bars) */
2023-06-13T03:01:14Z
<p><span dir="auto"><span class="autocomment">Solution 5 (Stars and Bars)</span></span></p>
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<td colspan="2" style="background-color: #fff; color: #222; text-align: center;">Revision as of 03:01, 13 June 2023</td>
</tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l110" >Line 110:</td>
<td colspan="2" class="diff-lineno">Line 110:</td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>Case 2: 1 pair</div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>Case 2: 1 pair</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>This is equivalent to the ant making 13 moves in the inner circle where it switches directions 2 times (counterclockwise-clockwise-counterclockwise). Note that the number of counterclockwise and clockwise moves <del class="diffchange diffchange-inline">have to </del>be <del class="diffchange diffchange-inline">are </del>equivalent<math>\pmod{5}</math>. Let the number of counterclockwise moves be <math>C</math> and clockwise moves be <math>C'</math>. The only two feasible solutions for <math>(C,C')</math> are <math>(4,9)</math> and <math>(9,4)</math>. Therefore, we use stars and bars to obtain <math>10+5=15</math> ways for this case.  </div></td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>This is equivalent to the ant making 13 moves in the inner circle where it switches directions 2 times (counterclockwise-clockwise-counterclockwise). Note that the number of counterclockwise and clockwise moves <ins class="diffchange diffchange-inline">must </ins>be equivalent<math>\pmod{5}</math>. Let the number of counterclockwise moves be <math>C</math> and clockwise moves be <math>C'</math>. The only two feasible solutions for <math>(C,C')</math> are <math>(4,9)</math> and <math>(9,4)</math>. Therefore, we use stars and bars to obtain <math>10+5=15</math> ways for this case.  </div></td></tr>
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<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>Case 3: 2 pairs</div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>Case 3: 2 pairs</div></td></tr>
</table>
Magnetoninja
https://artofproblemsolving.com/wiki/index.php?title=2018_AIME_I_Problems/Problem_10&diff=194309&oldid=prev
Magnetoninja: /* Solution 5 (Stars and Bars) */
2023-06-13T02:59:11Z
<p><span dir="auto"><span class="autocomment">Solution 5 (Stars and Bars)</span></span></p>
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<td colspan="2" style="background-color: #fff; color: #222; text-align: center;">Revision as of 02:59, 13 June 2023</td>
</tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l114" >Line 114:</td>
<td colspan="2" class="diff-lineno">Line 114:</td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>Case 3: 2 pairs</div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>Case 3: 2 pairs</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>The ant changes directions 4 times, so we obtain <math>(3,8)</math> and <math>(8,3)</math> using similar logic. Using stars and bars once again, we obtain <math>90+180=270</math> ways for this case.</div></td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>The ant changes directions 4 times <ins class="diffchange diffchange-inline">and makes 11 moves</ins>, so we obtain <math>(3,8)</math> and <math>(8,3)</math> using similar logic. Using stars and bars once again, we obtain <math>90+180=270</math> ways for this case.</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>Case 4: 3 pairs</div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>Case 4: 3 pairs</div></td></tr>
</table>
Magnetoninja
https://artofproblemsolving.com/wiki/index.php?title=2018_AIME_I_Problems/Problem_10&diff=194308&oldid=prev
Magnetoninja: /* Solution 5 (Stars and Bars) */
2023-06-13T02:57:24Z
<p><span dir="auto"><span class="autocomment">Solution 5 (Stars and Bars)</span></span></p>
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<td colspan="2" style="background-color: #fff; color: #222; text-align: center;">Revision as of 02:57, 13 June 2023</td>
</tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l130" >Line 130:</td>
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<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>The process of using stars and bars was the number of ways to group the counterclockwise and clockwise moves. For instance, if there are <math>p</math> pairs of switches, then based on the solutions for <math>(C,C')</math>, we can divide <math>C</math> into <math>p+1</math> subsets and <math>C'</math> into <math>p</math> subsets. <del class="diffchange diffchange-inline">Then, we multiply both results. We use the formula <math>\dbinom{a+b-1}{a-1}</math> for </del>putting <math><del class="diffchange diffchange-inline">b</del></math> objects <del class="diffchange diffchange-inline">into </del><math><del class="diffchange diffchange-inline">a</del></math> bins<del class="diffchange diffchange-inline">. Plugging in </del><math>C</math> <del class="diffchange diffchange-inline">for </del><math><del class="diffchange diffchange-inline">b</del></math> <del class="diffchange diffchange-inline">and </del><math><del class="diffchange diffchange-inline">C'</del></math> <del class="diffchange diffchange-inline">for </del><math><del class="diffchange diffchange-inline">a</del></math> <del class="diffchange diffchange-inline">gives us the values for the cases</del>.  </div></td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>The process of using stars and bars was the number of ways to group the counterclockwise and clockwise moves. For instance, if there are <math>p</math> pairs of switches, then based on the solutions for <math>(C,C')</math>, we can divide <math>C</math> into <math>p+1</math> subsets and <math>C'</math> into <math>p</math> subsets. <ins class="diffchange diffchange-inline">This is equivalent to </ins>putting <math><ins class="diffchange diffchange-inline">C</ins></math> objects <ins class="diffchange diffchange-inline">in </ins><math><ins class="diffchange diffchange-inline">p+1</ins></math> bins <ins class="diffchange diffchange-inline">and </ins><math>C<ins class="diffchange diffchange-inline">'</ins></math> <ins class="diffchange diffchange-inline">objects into </ins><math><ins class="diffchange diffchange-inline">p</ins></math> <ins class="diffchange diffchange-inline">bins where a bin can be empty which is modeled by consecutive switches. Then, we multiply both results. Therefore, for each </ins><math><ins class="diffchange diffchange-inline">p</ins></math><ins class="diffchange diffchange-inline">, we compute </ins><math><ins class="diffchange diffchange-inline">\dbinom{C+p}{p}\dbinom{C'+p-1}{p-1}</ins></math> <ins class="diffchange diffchange-inline">and add them up</ins>.</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>~[https://artofproblemsolving.com/wiki/index.php/User:Magnetoninja Magnetoninja]</div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>~[https://artofproblemsolving.com/wiki/index.php/User:Magnetoninja Magnetoninja]</div></td></tr>
</table>
Magnetoninja
https://artofproblemsolving.com/wiki/index.php?title=2018_AIME_I_Problems/Problem_10&diff=194295&oldid=prev
Magnetoninja: /* Solution 5 (Stars and Bars) */
2023-06-12T22:17:50Z
<p><span dir="auto"><span class="autocomment">Solution 5 (Stars and Bars)</span></span></p>
<table class="diff diff-contentalign-left" data-mw="interface">
<col class="diff-marker" />
<col class="diff-content" />
<col class="diff-marker" />
<col class="diff-content" />
<tr class="diff-title" lang="en">
<td colspan="2" style="background-color: #fff; color: #222; text-align: center;">← Older revision</td>
<td colspan="2" style="background-color: #fff; color: #222; text-align: center;">Revision as of 22:17, 12 June 2023</td>
</tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l130" >Line 130:</td>
<td colspan="2" class="diff-lineno">Line 130:</td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>The process of using stars and bars was the number of ways to group the counterclockwise and clockwise moves. For instance, if there are <math>p</math> <del class="diffchange diffchange-inline">"</del>pairs<del class="diffchange diffchange-inline">" </del>of switches, then based on the solutions for <math>(C,C')</math>, we can divide <math>C</math> into <math>p+1</math> subsets and <math>C'</math> into <math>p</math> subsets. Then, we multiply both results. We use the formula <math>\dbinom{a+b-1}{a-1}</math> for putting <math>b</math> objects into <math>a</math> bins. Plugging in <math>C</math> for <math>b</math> and <math>C'</math> for <math>a</math> gives us the values for the cases.  </div></td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>The process of using stars and bars was the number of ways to group the counterclockwise and clockwise moves. For instance, if there are <math>p</math> pairs of switches, then based on the solutions for <math>(C,C')</math>, we can divide <math>C</math> into <math>p+1</math> subsets and <math>C'</math> into <math>p</math> subsets. Then, we multiply both results. We use the formula <math>\dbinom{a+b-1}{a-1}</math> for putting <math>b</math> objects into <math>a</math> bins. Plugging in <math>C</math> for <math>b</math> and <math>C'</math> for <math>a</math> gives us the values for the cases.  </div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>~[https://artofproblemsolving.com/wiki/index.php/User:Magnetoninja Magnetoninja]</div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>~[https://artofproblemsolving.com/wiki/index.php/User:Magnetoninja Magnetoninja]</div></td></tr>
</table>
Magnetoninja
https://artofproblemsolving.com/wiki/index.php?title=2018_AIME_I_Problems/Problem_10&diff=194294&oldid=prev
Magnetoninja: /* Solution 5 (Stars and Bars) */
2023-06-12T22:17:30Z
<p><span dir="auto"><span class="autocomment">Solution 5 (Stars and Bars)</span></span></p>
<table class="diff diff-contentalign-left" data-mw="interface">
<col class="diff-marker" />
<col class="diff-content" />
<col class="diff-marker" />
<col class="diff-content" />
<tr class="diff-title" lang="en">
<td colspan="2" style="background-color: #fff; color: #222; text-align: center;">← Older revision</td>
<td colspan="2" style="background-color: #fff; color: #222; text-align: center;">Revision as of 22:17, 12 June 2023</td>
</tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l130" >Line 130:</td>
<td colspan="2" class="diff-lineno">Line 130:</td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>The process of using stars and bars was the number of ways to group the counterclockwise and clockwise moves. For instance, if there are <math>p</math> "pairs" of switches, then based on the solutions for <math>(C,C')</math>, we can <del class="diffchange diffchange-inline">partition </del><math>C</math> into <math>p+1</math> subsets and <math>C'</math> into <math>p</math> subsets. Then, we multiply both results. We use the formula <math>\dbinom{a+b-1}{a-1}</math> for putting <math>b</math> objects into <math>a</math> bins. Plugging in <math>C</math> for <math>b</math> and <math>C'</math> for <math>a</math> gives us the values for the cases.  </div></td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>The process of using stars and bars was the number of ways to group the counterclockwise and clockwise moves. For instance, if there are <math>p</math> "pairs" of switches, then based on the solutions for <math>(C,C')</math>, we can <ins class="diffchange diffchange-inline">divide </ins><math>C</math> into <math>p+1</math> subsets and <math>C'</math> into <math>p</math> subsets. Then, we multiply both results. We use the formula <math>\dbinom{a+b-1}{a-1}</math> for putting <math>b</math> objects into <math>a</math> bins. Plugging in <math>C</math> for <math>b</math> and <math>C'</math> for <math>a</math> gives us the values for the cases.  </div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>~[https://artofproblemsolving.com/wiki/index.php/User:Magnetoninja Magnetoninja]</div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>~[https://artofproblemsolving.com/wiki/index.php/User:Magnetoninja Magnetoninja]</div></td></tr>
</table>
Magnetoninja
https://artofproblemsolving.com/wiki/index.php?title=2018_AIME_I_Problems/Problem_10&diff=194293&oldid=prev
Magnetoninja: /* Solution 5 (Stars and Bars) */
2023-06-12T22:16:32Z
<p><span dir="auto"><span class="autocomment">Solution 5 (Stars and Bars)</span></span></p>
<table class="diff diff-contentalign-left" data-mw="interface">
<col class="diff-marker" />
<col class="diff-content" />
<col class="diff-marker" />
<col class="diff-content" />
<tr class="diff-title" lang="en">
<td colspan="2" style="background-color: #fff; color: #222; text-align: center;">← Older revision</td>
<td colspan="2" style="background-color: #fff; color: #222; text-align: center;">Revision as of 22:16, 12 June 2023</td>
</tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l110" >Line 110:</td>
<td colspan="2" class="diff-lineno">Line 110:</td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>Case 2: 1 pair</div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>Case 2: 1 pair</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>This is equivalent to the ant making 13 moves in the inner circle where it switches directions 2 times (counterclockwise-clockwise-counterclockwise). Note that the number of counterclockwise and clockwise moves are equivalent<math>\pmod{5}</math>. Let the number of counterclockwise moves be <math>C</math> and clockwise moves be <math>C'</math>. The only two feasible solutions for <math>(C,C')</math> are <math>(4,9)</math> and <math>(9,4)</math>. Therefore, we use stars and bars to obtain <math>10+5=15</math> ways for this case.  </div></td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>This is equivalent to the ant making 13 moves in the inner circle where it switches directions 2 times (counterclockwise-clockwise-counterclockwise). Note that the number of counterclockwise and clockwise moves <ins class="diffchange diffchange-inline">have to be </ins>are equivalent<math>\pmod{5}</math>. Let the number of counterclockwise moves be <math>C</math> and clockwise moves be <math>C'</math>. The only two feasible solutions for <math>(C,C')</math> are <math>(4,9)</math> and <math>(9,4)</math>. Therefore, we use stars and bars to obtain <math>10+5=15</math> ways for this case.  </div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>Case 3: 2 pairs</div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>Case 3: 2 pairs</div></td></tr>
</table>
Magnetoninja
https://artofproblemsolving.com/wiki/index.php?title=2018_AIME_I_Problems/Problem_10&diff=194292&oldid=prev
Magnetoninja: /* Solution 5 (Stars and Bars) */
2023-06-12T22:15:41Z
<p><span dir="auto"><span class="autocomment">Solution 5 (Stars and Bars)</span></span></p>
<table class="diff diff-contentalign-left" data-mw="interface">
<col class="diff-marker" />
<col class="diff-content" />
<col class="diff-marker" />
<col class="diff-content" />
<tr class="diff-title" lang="en">
<td colspan="2" style="background-color: #fff; color: #222; text-align: center;">← Older revision</td>
<td colspan="2" style="background-color: #fff; color: #222; text-align: center;">Revision as of 22:15, 12 June 2023</td>
</tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l130" >Line 130:</td>
<td colspan="2" class="diff-lineno">Line 130:</td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>The process of using stars and bars was the number of ways to group the counterclockwise and clockwise moves. For instance, if there are <math>p</math> "pairs" of switches, then based on the solutions for <math>(C,C')</math>, we can partition <math>C</math> into <math>p+1</math> subsets and <math>C'</math> into <math>p</math> subsets. Then, we multiply both results. We use the formula <math>\dbinom{a+b-1}{a-1}</math> for putting <math>b</math> objects into <math>a</math> bins. Plugging in <math>C</math> for <math>b</math> and <math>C'</math> for <math>a</math> gives us the values for the cases.</div></td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>The process of using stars and bars was the number of ways to group the counterclockwise and clockwise moves. For instance, if there are <math>p</math> "pairs" of switches, then based on the solutions for <math>(C,C')</math>, we can partition <math>C</math> into <math>p+1</math> subsets and <math>C'</math> into <math>p</math> subsets. Then, we multiply both results. We use the formula <math>\dbinom{a+b-1}{a-1}</math> for putting <math>b</math> objects into <math>a</math> bins. Plugging in <math>C</math> for <math>b</math> and <math>C'</math> for <math>a</math> gives us the values for the cases.  </div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>~[https://artofproblemsolving.com/wiki/index.php/User:Magnetoninja Magnetoninja]</div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>~[https://artofproblemsolving.com/wiki/index.php/User:Magnetoninja Magnetoninja]</div></td></tr>
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Magnetoninja
https://artofproblemsolving.com/wiki/index.php?title=2018_AIME_I_Problems/Problem_10&diff=194291&oldid=prev
Magnetoninja: /* Solution 5 (Stars and Bars) */
2023-06-12T22:14:56Z
<p><span dir="auto"><span class="autocomment">Solution 5 (Stars and Bars)</span></span></p>
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<td colspan="2" style="background-color: #fff; color: #222; text-align: center;">Revision as of 22:14, 12 June 2023</td>
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<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>Case 3: 2 pairs</div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>Case 3: 2 pairs</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del class="diffchange diffchange-inline">Using similar logic</del>, we obtain <math>(3,8)</math> and <math>(8,3)</math>. Using stars and bars once again, we obtain <math>90+180=270</math> ways for this case.</div></td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins class="diffchange diffchange-inline">The ant changes directions 4 times</ins>, <ins class="diffchange diffchange-inline">so </ins>we obtain <math>(3,8)</math> and <math>(8,3)</math> <ins class="diffchange diffchange-inline">using similar logic</ins>. Using stars and bars once again, we obtain <math>90+180=270</math> ways for this case.</div></td></tr>
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<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>Case 4: 3 pairs</div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>Case 4: 3 pairs</div></td></tr>
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Magnetoninja