https://artofproblemsolving.com/wiki/api.php?action=feedcontributions&user=Eatmathalive&feedformat=atom AoPS Wiki - User contributions [en] 2021-06-13T15:39:01Z User contributions MediaWiki 1.31.1 https://artofproblemsolving.com/wiki/index.php?title=2010_AMC_12A_Problems/Problem_18&diff=113341 2010 AMC 12A Problems/Problem 18 2019-12-24T03:46:29Z <p>Eatmathalive: /* Solution */</p> <hr /> <div>== Problem ==<br /> A 16-step path is to go from &lt;math&gt;(-4,-4)&lt;/math&gt; to &lt;math&gt;(4,4)&lt;/math&gt; with each step increasing either the &lt;math&gt;x&lt;/math&gt;-coordinate or the &lt;math&gt;y&lt;/math&gt;-coordinate by 1. How many such paths stay outside or on the boundary of the square &lt;math&gt;-2 \le x \le 2&lt;/math&gt;, &lt;math&gt;-2 \le y \le 2&lt;/math&gt; at each step?<br /> <br /> &lt;math&gt;\textbf{(A)}\ 92 \qquad \textbf{(B)}\ 144 \qquad \textbf{(C)}\ 1568 \qquad \textbf{(D)}\ 1698 \qquad \textbf{(E)}\ 12,800&lt;/math&gt;<br /> <br /> == Solution ==<br /> Each path must go through either the second or the fourth quadrant.<br /> Each path that goes through the second quadrant must pass through exactly one of the points &lt;math&gt;(-4,4)&lt;/math&gt;, &lt;math&gt;(-3,3)&lt;/math&gt;, and &lt;math&gt;(-2,2)&lt;/math&gt;.<br /> <br /> There is &lt;math&gt;1&lt;/math&gt; path of the first kind, &lt;math&gt;{8\choose 1}^2=64&lt;/math&gt; paths of the second kind, and &lt;math&gt;{8\choose 2}^2=28^2=784&lt;/math&gt; paths of the third type. <br /> Each path that goes through the fourth quadrant must pass through exactly one of the points &lt;math&gt;(4,-4)&lt;/math&gt;, &lt;math&gt;(3,-3)&lt;/math&gt;, and &lt;math&gt;(2,-2)&lt;/math&gt;.<br /> Again, there is &lt;math&gt;1&lt;/math&gt; path of the first kind, &lt;math&gt;{8\choose 1}^2=64&lt;/math&gt; paths of the second kind, and &lt;math&gt;{8\choose 2}^2=28^2=784&lt;/math&gt; paths of the third type. <br /> <br /> Hence the total number of paths is &lt;math&gt;2(1+64+784) = \boxed{1698}&lt;/math&gt;.<br /> <br /> Haha<br /> <br /> == See also ==<br /> {{AMC12 box|year=2010|num-b=17|num-a=19|ab=A}}<br /> <br /> [[Category:Introductory Combinatorics Problems]]<br /> {{MAA Notice}}</div> Eatmathalive