Difference between revisions of "2021 AMC 10A Problems/Problem 20"
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<math>\textbf{(A)} ~10\qquad\textbf{(B)} ~18\qquad\textbf{(C)} ~24 \qquad\textbf{(D)} ~32 \qquad\textbf{(E)} ~44</math> | <math>\textbf{(A)} ~10\qquad\textbf{(B)} ~18\qquad\textbf{(C)} ~24 \qquad\textbf{(D)} ~32 \qquad\textbf{(E)} ~44</math> | ||
− | ==Solution 1 ( | + | ==Solution 1 (Enumeration)== |
− | We write out the <math>120</math> cases | + | We write out the <math>5!=120</math> cases, then filter out the valid ones: |
− | + | ||
<math>13254,14253,14352,15243,15342,21435,21534,23154,24153,24351,25143,25341,\linebreak | <math>13254,14253,14352,15243,15342,21435,21534,23154,24153,24351,25143,25341,\linebreak | ||
− | 31425,31524,32415, | + | 31425,31524,32415,32514,34152,34251,35142,35241,41325,41523,42315,42513,\linebreak |
− | 43512,45132,45231,51324,51423,52314,52413,53412. | + | 43512,45132,45231,51324,51423,52314,52413,53412.</math> |
− | We count these out and get <math>\boxed{\ | + | |
+ | We count these out and get <math>\boxed{\textbf{(D)} ~32}</math> permutations that work. | ||
+ | |||
+ | ~contactbibliophile | ||
− | ==Solution 2 (Casework on the Consecutive Digits)== | + | ==Solution 2 (Enumeration by Symmetry)== |
+ | By symmetry with respect to <math>3,</math> note that <math>(x_1,x_2,x_3,x_4,x_5)</math> is a valid sequence if and only if <math>(6-x_1,6-x_2,6-x_3,6-x_4,6-x_5)</math> is a valid sequence. We enumerate the valid sequences that start with <math>1,2,31,</math> or <math>32,</math> as shown below: | ||
+ | |||
+ | <asy> | ||
+ | /* Made by MRENTHUSIASM */ | ||
+ | size(15cm); | ||
+ | |||
+ | draw((0.25,0)--(1.75,3),red,EndArrow); | ||
+ | draw((0.25,0)--(1.75,0),red,EndArrow); | ||
+ | draw((0.25,0)--(1.75,-3),red,EndArrow); | ||
+ | draw((2.25,3)--(3.75,3),red,EndArrow); | ||
+ | draw((2.25,0)--(3.75,3/4),red,EndArrow); | ||
+ | draw((2.25,0)--(3.75,-3/4),red,EndArrow); | ||
+ | draw((2.25,-3)--(3.75,-9/4),red,EndArrow); | ||
+ | draw((2.25,-3)--(3.75,-15/4),red,EndArrow); | ||
+ | draw((4.25,3)--(5.75,3),red,EndArrow); | ||
+ | draw((4.25,3/4)--(5.75,3/4),red,EndArrow); | ||
+ | draw((4.25,-3/4)--(5.75,-3/4),red,EndArrow); | ||
+ | draw((4.25,-9/4)--(5.75,-9/4),red,EndArrow); | ||
+ | draw((4.25,-15/4)--(5.75,-15/4),red,EndArrow); | ||
+ | draw((6.25,3)--(7.75,3),red,EndArrow); | ||
+ | draw((6.25,3/4)--(7.75,3/4),red,EndArrow); | ||
+ | draw((6.25,-3/4)--(7.75,-3/4),red,EndArrow); | ||
+ | draw((6.25,-9/4)--(7.75,-9/4),red,EndArrow); | ||
+ | draw((6.25,-15/4)--(7.75,-15/4),red,EndArrow); | ||
+ | |||
+ | label("$1$",(0,0)); | ||
+ | label("$3$",(2,3)); | ||
+ | label("$2$",(4,3)); | ||
+ | label("$5$",(6,3)); | ||
+ | label("$4$",(8,3)); | ||
+ | |||
+ | label("$4$",(2,0)); | ||
+ | label("$2$",(4,3/4)); | ||
+ | label("$5$",(6,3/4)); | ||
+ | label("$3$",(8,3/4)); | ||
+ | label("$3$",(4,-3/4)); | ||
+ | label("$5$",(6,-3/4)); | ||
+ | label("$2$",(8,-3/4)); | ||
+ | |||
+ | label("$5$",(2,-3)); | ||
+ | label("$2$",(4,-9/4)); | ||
+ | label("$4$",(6,-9/4)); | ||
+ | label("$3$",(8,-9/4)); | ||
+ | label("$3$",(4,-15/4)); | ||
+ | label("$4$",(6,-15/4)); | ||
+ | label("$2$",(8,-15/4)); | ||
+ | |||
+ | draw((0.25,-10.5-3/4)--(1.75,-6-3/4),red,EndArrow); | ||
+ | draw((0.25,-10.5-3/4)--(1.75,-9-3/4),red,EndArrow); | ||
+ | draw((0.25,-10.5-3/4)--(1.75,-12-3/4),red,EndArrow); | ||
+ | draw((0.25,-10.5-3/4)--(1.75,-15-3/4),red,EndArrow); | ||
+ | draw((2.25,-6-3/4)--(3.75,-6),red,EndArrow); | ||
+ | draw((2.25,-6-3/4)--(3.75,-6-6/4),red,EndArrow); | ||
+ | draw((2.25,-9-3/4)--(3.75,-9-3/4),red,EndArrow); | ||
+ | draw((2.25,-12-3/4)--(3.75,-12),red,EndArrow); | ||
+ | draw((2.25,-12-3/4)--(3.75,-12-6/4),red,EndArrow); | ||
+ | draw((2.25,-15-3/4)--(3.75,-15),red,EndArrow); | ||
+ | draw((2.25,-15-3/4)--(3.75,-15-6/4),red,EndArrow); | ||
+ | draw((4.25,-6)--(5.75,-6),red,EndArrow); | ||
+ | draw((4.25,-6-6/4)--(5.75,-6-6/4),red,EndArrow); | ||
+ | draw((4.25,-9-3/4)--(5.75,-9-3/4),red,EndArrow); | ||
+ | draw((4.25,-12)--(5.75,-12),red,EndArrow); | ||
+ | draw((4.25,-12-6/4)--(5.75,-12-6/4),red,EndArrow); | ||
+ | draw((4.25,-15)--(5.75,-15),red,EndArrow); | ||
+ | draw((4.25,-15-6/4)--(5.75,-15-6/4),red,EndArrow); | ||
+ | draw((6.25,-6)--(7.75,-6),red,EndArrow); | ||
+ | draw((6.25,-6-6/4)--(7.75,-6-6/4),red,EndArrow); | ||
+ | draw((6.25,-9-3/4)--(7.75,-9-3/4),red,EndArrow); | ||
+ | draw((6.25,-12)--(7.75,-12),red,EndArrow); | ||
+ | draw((6.25,-12-6/4)--(7.75,-12-6/4),red,EndArrow); | ||
+ | draw((6.25,-15)--(7.75,-15),red,EndArrow); | ||
+ | draw((6.25,-15-6/4)--(7.75,-15-6/4),red,EndArrow); | ||
+ | |||
+ | label("$2$",(0,-10.5-3/4)); | ||
+ | label("$1$",(2,-6-3/4)); | ||
+ | label("$3$",(2,-9-3/4)); | ||
+ | label("$4$",(2,-12-3/4)); | ||
+ | label("$5$",(2,-15-3/4)); | ||
+ | |||
+ | label("$4$",(4,-6)); | ||
+ | label("$5$",(4,-6-6/4)); | ||
+ | label("$1$",(4,-9-3/4)); | ||
+ | label("$1$",(4,-12)); | ||
+ | label("$3$",(4,-12-6/4)); | ||
+ | label("$1$",(4,-15)); | ||
+ | label("$3$",(4,-15-6/4)); | ||
+ | |||
+ | label("$3$",(6,-6)); | ||
+ | label("$3$",(6,-6-6/4)); | ||
+ | label("$5$",(6,-9-3/4)); | ||
+ | label("$5$",(6,-12)); | ||
+ | label("$5$",(6,-12-6/4)); | ||
+ | label("$4$",(6,-15)); | ||
+ | label("$4$",(6,-15-6/4)); | ||
+ | |||
+ | label("$5$",(8,-6)); | ||
+ | label("$4$",(8,-6-6/4)); | ||
+ | label("$4$",(8,-9-3/4)); | ||
+ | label("$3$",(8,-12)); | ||
+ | label("$1$",(8,-12-6/4)); | ||
+ | label("$3$",(8,-15)); | ||
+ | label("$1$",(8,-15-6/4)); | ||
+ | |||
+ | draw((0.25,-7.5-3/4-12.75)--(1.75,-6-3/4-12.75),red,EndArrow); | ||
+ | draw((0.25,-7.5-3/4-12.75)--(1.75,-9-3/4-12.75),red,EndArrow); | ||
+ | draw((2.25,-6-3/4-12.75)--(3.75,-6-12.75),red,EndArrow); | ||
+ | draw((2.25,-6-3/4-12.75)--(3.75,-6-6/4-12.75),red,EndArrow); | ||
+ | draw((2.25,-9-3/4-12.75)--(3.75,-9-12.75),red,EndArrow); | ||
+ | draw((2.25,-9-3/4-12.75)--(3.75,-9-6/4-12.75),red,EndArrow); | ||
+ | draw((4.25,-6-12.75)--(5.75,-6-12.75),red,EndArrow); | ||
+ | draw((4.25,-6-6/4-12.75)--(5.75,-6-6/4-12.75),red,EndArrow); | ||
+ | draw((4.25,-9-12.75)--(5.75,-9-12.75),red,EndArrow); | ||
+ | draw((4.25,-9-6/4-12.75)--(5.75,-9-6/4-12.75),red,EndArrow); | ||
+ | draw((6.25,-6-12.75)--(7.75,-6-12.75),red,EndArrow); | ||
+ | draw((6.25,-6-6/4-12.75)--(7.75,-6-6/4-12.75),red,EndArrow); | ||
+ | draw((6.25,-9-12.75)--(7.75,-9-12.75),red,EndArrow); | ||
+ | draw((6.25,-9-6/4-12.75)--(7.75,-9-6/4-12.75),red,EndArrow); | ||
+ | |||
+ | label("$3$",(0,-7.5-3/4-12.75)); | ||
+ | label("$1$",(2,-6-3/4-12.75)); | ||
+ | label("$2$",(2,-9-3/4-12.75)); | ||
+ | |||
+ | label("$4$",(4,-6-12.75)); | ||
+ | label("$5$",(4,-6-6/4-12.75)); | ||
+ | label("$4$",(4,-9-12.75)); | ||
+ | label("$5$",(4,-9-6/4-12.75)); | ||
+ | |||
+ | label("$2$",(6,-6-12.75)); | ||
+ | label("$2$",(6,-6-6/4-12.75)); | ||
+ | label("$1$",(6,-9-12.75)); | ||
+ | label("$1$",(6,-9-6/4-12.75)); | ||
+ | |||
+ | label("$5$",(8,-6-12.75)); | ||
+ | label("$4$",(8,-6-6/4-12.75)); | ||
+ | label("$5$",(8,-9-12.75)); | ||
+ | label("$4$",(8,-9-6/4-12.75)); | ||
+ | </asy> | ||
+ | |||
+ | There are <math>16</math> valid sequences that start with <math>1,2,31,</math> or <math>32.</math> By symmetry, there are <math>16</math> valid sequences that start with <math>5,4,35,</math> or <math>34.</math> So, the answer is <math>16+16=\boxed{\textbf{(D)} ~32}.</math> | ||
+ | |||
+ | ~MRENTHUSIASM (inspired by Snowfan) | ||
+ | |||
+ | ==Solution 3 (Casework on the Consecutive Digits)== | ||
Reading the terms from left to right, we have two cases for the consecutive digits, where <math>+</math> means increase and <math>-</math> means decrease: | Reading the terms from left to right, we have two cases for the consecutive digits, where <math>+</math> means increase and <math>-</math> means decrease: | ||
Line 19: | Line 165: | ||
<math>\textbf{Case \#2: }\boldsymbol{-,+,-,+}</math> | <math>\textbf{Case \#2: }\boldsymbol{-,+,-,+}</math> | ||
− | For <math>\text{Case \#1},</math> note that for the second and fourth terms, one term must be <math>5,</math> and the other term must be <math>3</math> or <math>4.</math> We have four | + | For <math>\text{Case \#1},</math> note that for the second and fourth terms, one term must be <math>5,</math> and the other term must be either <math>3</math> or <math>4.</math> We have four subcases: |
<math>(1) \ \underline{\hspace{3mm}}3\underline{\hspace{3mm}}5\underline{\hspace{3mm}}</math> | <math>(1) \ \underline{\hspace{3mm}}3\underline{\hspace{3mm}}5\underline{\hspace{3mm}}</math> | ||
Line 29: | Line 175: | ||
<math>(4) \ \underline{\hspace{3mm}}5\underline{\hspace{3mm}}4\underline{\hspace{3mm}}</math> | <math>(4) \ \underline{\hspace{3mm}}5\underline{\hspace{3mm}}4\underline{\hspace{3mm}}</math> | ||
− | For <math>(1),</math> the first two blanks must be <math>1</math> and <math>2</math> in some order, and the last blank must be <math>4 | + | For <math>(1),</math> the first two blanks must be <math>1</math> and <math>2</math> in some order, and the last blank must be <math>4.</math> So, we get <math>2</math> possibilities. Similarly, <math>(2)</math> also has <math>2</math> possibilities. |
− | For <math>(3),</math> there are no restrictions for the numbers <math>1, 2,</math> and <math>3.</math> So, we | + | For <math>(3),</math> there are no restrictions for the numbers <math>1, 2,</math> and <math>3.</math> So, we get <math>3!=6</math> possibilities. Similarly, <math>(4)</math> also has <math>6</math> possibilities. |
Together, <math>\text{Case \#1}</math> has <math>2+2+6+6=16</math> possibilities. By symmetry, <math>\text{Case \#2}</math> also has <math>16</math> possibilities. | Together, <math>\text{Case \#1}</math> has <math>2+2+6+6=16</math> possibilities. By symmetry, <math>\text{Case \#2}</math> also has <math>16</math> possibilities. | ||
Line 37: | Line 183: | ||
Finally, the answer is <math>16+16=\boxed{\textbf{(D)} ~32}.</math> | Finally, the answer is <math>16+16=\boxed{\textbf{(D)} ~32}.</math> | ||
− | This problem is | + | <u><b>Remark</b></u> |
+ | |||
+ | This problem is somewhat similar to [[2004_AIME_I_Problems/Problem_6|2004 AIME I Problem 6]]. | ||
~MRENTHUSIASM | ~MRENTHUSIASM | ||
− | ==Solution | + | ==Solution 4 (Casework Similar to Solution 3)== |
− | Like Solution | + | Like Solution 3, we have two cases. Due to symmetry, we just need to count one of the cases. For the purpose of this solution, we will be doing <math>-,+,-,+</math>. Instead of starting with 5, we start with 1. |
There are two ways to place it: | There are two ways to place it: | ||
Line 60: | Line 208: | ||
For case 2, we need to consider how many ways to arrange 3,4,5 in a a>b<c fashion. It should seem pretty obvious that b has to be 3, so there will be 2! way to put 4 and 5. | For case 2, we need to consider how many ways to arrange 3,4,5 in a a>b<c fashion. It should seem pretty obvious that b has to be 3, so there will be 2! way to put 4 and 5. | ||
− | Now we find our result, times 2 for symmetry, times 2 for placement of 1 and times (3!+2!) for the two different cases for placement of 2. This give us <math>2*2*(3!+2!)=4*(6+2)=32</math>. | + | Now we find our result, times 2 for symmetry, times 2 for placement of 1 and times (3!+2!) for the two different cases for placement of 2. This give us <math>2*2*(3!+2!)=4*(6+2)=\boxed{\textbf{(D)} ~32}</math>. |
~~Xhte | ~~Xhte | ||
− | == Solution | + | == Solution 5 (Casework on the Position of 5) == |
We only need to find the # of rearrangements when 5 is the 4th digit and 5th digit. Find the total, and multiply by 2. Then we can get the answer by adding the case when 5 is the third digit. | We only need to find the # of rearrangements when 5 is the 4th digit and 5th digit. Find the total, and multiply by 2. Then we can get the answer by adding the case when 5 is the third digit. | ||
− | |||
'''Case <math>1</math>:''' 5 is the 5th digit. __ __ __ __ 5 | '''Case <math>1</math>:''' 5 is the 5th digit. __ __ __ __ 5 | ||
Line 76: | Line 223: | ||
__ __ 4 __ 5, then the 1st digit must be <math>2</math> or <math>3</math>, <math>2</math> gives us <math>1</math> way, and <math>3</math> gives us <math>2</math> ways. (Can't be <math>1</math> because the first digit would increasing). Therefore, <math>4</math> in the middle and <math>5</math> in the last would result in <math>3</math> ways. | __ __ 4 __ 5, then the 1st digit must be <math>2</math> or <math>3</math>, <math>2</math> gives us <math>1</math> way, and <math>3</math> gives us <math>2</math> ways. (Can't be <math>1</math> because the first digit would increasing). Therefore, <math>4</math> in the middle and <math>5</math> in the last would result in <math>3</math> ways. | ||
− | |||
'''Case <math>2</math>:''' <math>5</math> is the fourth digit. __ __ __ 5 __ | '''Case <math>2</math>:''' <math>5</math> is the fourth digit. __ __ __ 5 __ | ||
Then the last digit can be all of the 4 numbers <math>1</math>, <math>2</math>, <math>3</math>, and <math>4</math>. Let's say if the last digit is <math>4</math>, then the 2nd digit would be the largest for the remaining digits to prevent increasing order or decreasing order. Then the remaining two are interchangeable, give us <math>2!</math> ways. All of the <math>4</math> can work, so case <math>2</math> would result in <math>2!+2!+2!+2!=8</math> ways. | Then the last digit can be all of the 4 numbers <math>1</math>, <math>2</math>, <math>3</math>, and <math>4</math>. Let's say if the last digit is <math>4</math>, then the 2nd digit would be the largest for the remaining digits to prevent increasing order or decreasing order. Then the remaining two are interchangeable, give us <math>2!</math> ways. All of the <math>4</math> can work, so case <math>2</math> would result in <math>2!+2!+2!+2!=8</math> ways. | ||
− | |||
'''Case <math>3</math>:''' <math>5</math> is in the middle. __ __ 5 __ __ | '''Case <math>3</math>:''' <math>5</math> is in the middle. __ __ 5 __ __ | ||
Line 92: | Line 237: | ||
<math>8+3+2=13</math>, so the total ways for case 1 and case 2 with both increasing and decreasing would be <math>13*2=26.</math> | <math>8+3+2=13</math>, so the total ways for case 1 and case 2 with both increasing and decreasing would be <math>13*2=26.</math> | ||
− | <math>26+6=\boxed{\textbf{(D)} ~32}.</math> | + | Finally, we have <math>26+6=\boxed{\textbf{(D)} ~32}.</math> |
~Michael595 | ~Michael595 |
Latest revision as of 22:00, 29 July 2021
Contents
- 1 Problem
- 2 Solution 1 (Enumeration)
- 3 Solution 2 (Enumeration by Symmetry)
- 4 Solution 3 (Casework on the Consecutive Digits)
- 5 Solution 4 (Casework Similar to Solution 3)
- 6 Solution 5 (Casework on the Position of 5)
- 7 Video Solution by OmegaLearn (Using PIE - Principle of Inclusion Exclusion)
- 8 Video Solution by Power of Logic (Using Idea of Symmetrically Counting)
- 9 Video Solution by TheBeautyofMath
- 10 See Also
Problem
In how many ways can the sequence be rearranged so that no three consecutive terms are increasing and no three consecutive terms are decreasing?
Solution 1 (Enumeration)
We write out the cases, then filter out the valid ones:
We count these out and get permutations that work.
~contactbibliophile
Solution 2 (Enumeration by Symmetry)
By symmetry with respect to note that is a valid sequence if and only if is a valid sequence. We enumerate the valid sequences that start with or as shown below:
There are valid sequences that start with or By symmetry, there are valid sequences that start with or So, the answer is
~MRENTHUSIASM (inspired by Snowfan)
Solution 3 (Casework on the Consecutive Digits)
Reading the terms from left to right, we have two cases for the consecutive digits, where means increase and means decrease:
For note that for the second and fourth terms, one term must be and the other term must be either or We have four subcases:
For the first two blanks must be and in some order, and the last blank must be So, we get possibilities. Similarly, also has possibilities.
For there are no restrictions for the numbers and So, we get possibilities. Similarly, also has possibilities.
Together, has possibilities. By symmetry, also has possibilities.
Finally, the answer is
Remark
This problem is somewhat similar to 2004 AIME I Problem 6.
~MRENTHUSIASM
Solution 4 (Casework Similar to Solution 3)
Like Solution 3, we have two cases. Due to symmetry, we just need to count one of the cases. For the purpose of this solution, we will be doing . Instead of starting with 5, we start with 1.
There are two ways to place it:
_1_ _ _
_ _ _1_
Now we place 2, it can either be next to 1 and on the outside, or is place in where 1 would go in the other case. So now we have another two "sub case":
_1_2_(case 1)
21_ _ _(case 2)
There are 3! ways to arrange the rest for case 1, since there is no restriction.
For case 2, we need to consider how many ways to arrange 3,4,5 in a a>b<c fashion. It should seem pretty obvious that b has to be 3, so there will be 2! way to put 4 and 5.
Now we find our result, times 2 for symmetry, times 2 for placement of 1 and times (3!+2!) for the two different cases for placement of 2. This give us .
~~Xhte
Solution 5 (Casework on the Position of 5)
We only need to find the # of rearrangements when 5 is the 4th digit and 5th digit. Find the total, and multiply by 2. Then we can get the answer by adding the case when 5 is the third digit.
Case : 5 is the 5th digit. __ __ __ __ 5
Then can only be either 1st digit or the 3rd digit.
4 __ __ __ 5, then the only way is that is the 3rd digit, so it can be either or , give us results.
__ __ 4 __ 5, then the 1st digit must be or , gives us way, and gives us ways. (Can't be because the first digit would increasing). Therefore, in the middle and in the last would result in ways.
Case : is the fourth digit. __ __ __ 5 __
Then the last digit can be all of the 4 numbers , , , and . Let's say if the last digit is , then the 2nd digit would be the largest for the remaining digits to prevent increasing order or decreasing order. Then the remaining two are interchangeable, give us ways. All of the can work, so case would result in ways.
Case : is in the middle. __ __ 5 __ __
Then there are only two cases: 1. , then 4 and 3 are interchangeable, which results in . Or it can be , then 4 and 2 are interchangeable, but it can not be , so there can only be 2 possible ways: , .
Therefore, case 3 would result in ways.
, so the total ways for case 1 and case 2 with both increasing and decreasing would be
Finally, we have
~Michael595
Video Solution by OmegaLearn (Using PIE - Principle of Inclusion Exclusion)
~ pi_is_3.14
Video Solution by Power of Logic (Using Idea of Symmetrically Counting)
Video Solution by TheBeautyofMath
~IceMatrix
See Also
2021 AMC 10A (Problems • Answer Key • Resources) | ||
Preceded by Problem 19 |
Followed by Problem 21 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | ||
All AMC 10 Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.