Difference between revisions of "2021 AMC 10A Problems/Problem 25"
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MRENTHUSIASM (talk | contribs) m (→Solution 4 (Casework on the Red Chips' Configurations)) |
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-happykeeper | -happykeeper | ||
− | ==Solution 2 (Casework)== | + | ==Solution 2 (Casework on the Top-Center and Center-Left Chips)== |
− | Without the loss of generality, we | + | Without the loss of generality, we fix the top-left square with a red chip. We apply casework to its two adjacent chips: |
− | <u><b>Case (1): The | + | <u><b>Case (1): The top-center and center-left chips have different colors.</b></u> |
− | < | + | <asy> |
− | + | /* Made by MRENTHUSIASM */ | |
− | + | unitsize(7mm); | |
− | + | fill((6,2)--(7,2)--(7,3)--(6,3)--cycle, red); | |
− | + | fill((7,2)--(8,2)--(8,3)--(7,3)--cycle, blue); | |
− | + | fill((6,1)--(7,1)--(7,2)--(6,2)--cycle, green); | |
− | + | fill((7,1)--(8,1)--(8,2)--(7,2)--cycle, red); | |
− | + | draw((6,0)--(9,0)--(9,3)--(6,3)--cycle, linewidth(1.5)); | |
− | + | draw((6,1)--(9,1), linewidth(1.5)); | |
− | + | draw((6,2)--(9,2), linewidth(1.5)); | |
− | + | draw((7,0)--(7,3), linewidth(1.5)); | |
− | + | draw((8,0)--(8,3), linewidth(1.5)); | |
+ | </asy> | ||
+ | There are three sub-cases for Case (1): | ||
+ | <asy> | ||
+ | /* Made by MRENTHUSIASM */ | ||
+ | unitsize(7mm); | ||
+ | fill((0,2)--(1,2)--(1,3)--(0,3)--cycle, red); | ||
+ | fill((1,2)--(2,2)--(2,3)--(1,3)--cycle, blue); | ||
+ | fill((0,1)--(1,1)--(1,2)--(0,2)--cycle, green); | ||
+ | fill((1,1)--(2,1)--(2,2)--(1,2)--cycle, red); | ||
+ | fill((2,2)--(3,2)--(3,3)--(2,3)--cycle, red); | ||
+ | fill((2,1)--(3,1)--(3,2)--(2,2)--cycle, green); | ||
+ | fill((2,0)--(3,0)--(3,1)--(2,1)--cycle, blue); | ||
+ | fill((0,0)--(1,0)--(1,1)--(0,1)--cycle, blue); | ||
+ | fill((1,0)--(2,0)--(2,1)--(1,1)--cycle, green); | ||
+ | |||
+ | fill((6,2)--(7,2)--(7,3)--(6,3)--cycle, red); | ||
+ | fill((7,2)--(8,2)--(8,3)--(7,3)--cycle, blue); | ||
+ | fill((6,1)--(7,1)--(7,2)--(6,2)--cycle, green); | ||
+ | fill((7,1)--(8,1)--(8,2)--(7,2)--cycle, red); | ||
+ | fill((8,2)--(9,2)--(9,3)--(8,3)--cycle, green); | ||
+ | fill((8,1)--(9,1)--(9,2)--(8,2)--cycle, blue); | ||
+ | fill((8,0)--(9,0)--(9,1)--(8,1)--cycle, green); | ||
+ | fill((6,0)--(7,0)--(7,1)--(6,1)--cycle, red); | ||
+ | fill((7,0)--(8,0)--(8,1)--(7,1)--cycle, blue); | ||
+ | |||
+ | fill((12,2)--(13,2)--(13,3)--(12,3)--cycle, red); | ||
+ | fill((13,2)--(14,2)--(14,3)--(13,3)--cycle, blue); | ||
+ | fill((12,1)--(13,1)--(13,2)--(12,2)--cycle, green); | ||
+ | fill((13,1)--(14,1)--(14,2)--(13,2)--cycle, red); | ||
+ | fill((14,2)--(15,2)--(15,3)--(14,3)--cycle, green); | ||
+ | fill((14,1)--(15,1)--(15,2)--(14,2)--cycle, blue); | ||
+ | fill((14,0)--(15,0)--(15,1)--(14,1)--cycle, red); | ||
+ | fill((12,0)--(13,0)--(13,1)--(12,1)--cycle, blue); | ||
+ | fill((13,0)--(14,0)--(14,1)--(13,1)--cycle, green); | ||
+ | |||
+ | draw((0,0)--(3,0)--(3,3)--(0,3)--cycle, linewidth(1.5)); | ||
+ | draw((0,1)--(3,1), linewidth(1.5)); | ||
+ | draw((0,2)--(3,2), linewidth(1.5)); | ||
+ | draw((1,0)--(1,3), linewidth(1.5)); | ||
+ | draw((2,0)--(2,3), linewidth(1.5)); | ||
− | + | draw((6,0)--(9,0)--(9,3)--(6,3)--cycle, linewidth(1.5)); | |
− | + | draw((6,1)--(9,1), linewidth(1.5)); | |
− | + | draw((6,2)--(9,2), linewidth(1.5)); | |
− | + | draw((7,0)--(7,3), linewidth(1.5)); | |
− | + | draw((8,0)--(8,3), linewidth(1.5)); | |
− | + | ||
− | + | draw((12,0)--(15,0)--(15,3)--(12,3)--cycle, linewidth(1.5)); | |
− | + | draw((12,1)--(15,1), linewidth(1.5)); | |
− | + | draw((12,2)--(15,2), linewidth(1.5)); | |
− | + | draw((13,0)--(13,3), linewidth(1.5)); | |
− | + | draw((14,0)--(14,3), linewidth(1.5)); | |
− | + | </asy> | |
− | + | As there are <math>3!=6</math> permutations of the three colors, each sub-case has <math>6</math> ways. So, Case (1) has <math>3\cdot6=18</math> ways in total. | |
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− | So, Case (1) has <math>3\cdot6=18</math> ways. | ||
− | <u><b>Case (2): The | + | <u><b>Case (2): The top-center and center-left chips have the same color.</b></u> |
− | < | + | <asy> |
− | + | /* Made by MRENTHUSIASM */ | |
− | + | unitsize(7mm); | |
− | + | fill((6,2)--(7,2)--(7,3)--(6,3)--cycle, red); | |
− | + | fill((7,2)--(8,2)--(8,3)--(7,3)--cycle, blue); | |
− | + | fill((6,1)--(7,1)--(7,2)--(6,2)--cycle, blue); | |
− | + | draw((6,0)--(9,0)--(9,3)--(6,3)--cycle, linewidth(1.5)); | |
− | + | draw((6,1)--(9,1), linewidth(1.5)); | |
− | + | draw((6,2)--(9,2), linewidth(1.5)); | |
− | + | draw((7,0)--(7,3), linewidth(1.5)); | |
+ | draw((8,0)--(8,3), linewidth(1.5)); | ||
+ | </asy> | ||
There are three sub-cases for Case (2): | There are three sub-cases for Case (2): | ||
− | < | + | <asy> |
− | + | /* Made by MRENTHUSIASM */ | |
− | + | unitsize(7mm); | |
− | + | fill((0,2)--(1,2)--(1,3)--(0,3)--cycle, red); | |
− | + | fill((1,2)--(2,2)--(2,3)--(1,3)--cycle, blue); | |
− | + | fill((0,1)--(1,1)--(1,2)--(0,2)--cycle, blue); | |
− | + | fill((1,1)--(2,1)--(2,2)--(1,2)--cycle, green); | |
− | + | fill((2,2)--(3,2)--(3,3)--(2,3)--cycle, red); | |
− | + | fill((2,1)--(3,1)--(3,2)--(2,2)--cycle, blue); | |
− | + | fill((2,0)--(3,0)--(3,1)--(2,1)--cycle, green); | |
− | + | fill((0,0)--(1,0)--(1,1)--(0,1)--cycle, green); | |
− | + | fill((1,0)--(2,0)--(2,1)--(1,1)--cycle, red); | |
− | + | ||
− | + | fill((6,2)--(7,2)--(7,3)--(6,3)--cycle, red); | |
− | + | fill((7,2)--(8,2)--(8,3)--(7,3)--cycle, blue); | |
− | + | fill((6,1)--(7,1)--(7,2)--(6,2)--cycle, blue); | |
− | + | fill((7,1)--(8,1)--(8,2)--(7,2)--cycle, green); | |
− | + | fill((8,2)--(9,2)--(9,3)--(8,3)--cycle, green); | |
− | + | fill((8,1)--(9,1)--(9,2)--(8,2)--cycle, red); | |
− | + | fill((8,0)--(9,0)--(9,1)--(8,1)--cycle, green); | |
− | + | fill((6,0)--(7,0)--(7,1)--(6,1)--cycle, red); | |
− | + | fill((7,0)--(8,0)--(8,1)--(7,1)--cycle, blue); | |
− | + | ||
− | + | fill((12,2)--(13,2)--(13,3)--(12,3)--cycle, red); | |
− | + | fill((13,2)--(14,2)--(14,3)--(13,3)--cycle, blue); | |
− | + | fill((12,1)--(13,1)--(13,2)--(12,2)--cycle, blue); | |
− | + | fill((13,1)--(14,1)--(14,2)--(13,2)--cycle, green); | |
− | + | fill((14,2)--(15,2)--(15,3)--(14,3)--cycle, green); | |
− | + | fill((14,1)--(15,1)--(15,2)--(14,2)--cycle, red); | |
− | + | fill((14,0)--(15,0)--(15,1)--(14,1)--cycle, blue); | |
− | + | fill((12,0)--(13,0)--(13,1)--(12,1)--cycle, green); | |
− | So, Case (2) has <math>3\cdot6=18</math> ways. | + | fill((13,0)--(14,0)--(14,1)--(13,1)--cycle, red); |
+ | |||
+ | draw((0,0)--(3,0)--(3,3)--(0,3)--cycle, linewidth(1.5)); | ||
+ | draw((0,1)--(3,1), linewidth(1.5)); | ||
+ | draw((0,2)--(3,2), linewidth(1.5)); | ||
+ | draw((1,0)--(1,3), linewidth(1.5)); | ||
+ | draw((2,0)--(2,3), linewidth(1.5)); | ||
+ | |||
+ | draw((6,0)--(9,0)--(9,3)--(6,3)--cycle, linewidth(1.5)); | ||
+ | draw((6,1)--(9,1), linewidth(1.5)); | ||
+ | draw((6,2)--(9,2), linewidth(1.5)); | ||
+ | draw((7,0)--(7,3), linewidth(1.5)); | ||
+ | draw((8,0)--(8,3), linewidth(1.5)); | ||
+ | |||
+ | draw((12,0)--(15,0)--(15,3)--(12,3)--cycle, linewidth(1.5)); | ||
+ | draw((12,1)--(15,1), linewidth(1.5)); | ||
+ | draw((12,2)--(15,2), linewidth(1.5)); | ||
+ | draw((13,0)--(13,3), linewidth(1.5)); | ||
+ | draw((14,0)--(14,3), linewidth(1.5)); | ||
+ | </asy> | ||
+ | As there are <math>3!=6</math> permutations of the three colors, each sub-case has <math>6</math> ways. So, Case (2) has <math>3\cdot6=18</math> ways in total. | ||
<u><b>Answer</b></u> | <u><b>Answer</b></u> | ||
Line 157: | Line 190: | ||
− | <b>Case (2)</b>: All of the rows have two | + | <b>Case (2)</b>: All of the rows have two chips that are the same color and one that is different. There are obviously <math>3</math> possible configurations for the first row, <math>2</math> for the second, and <math>2</math> for the third. <math>3\cdot2\cdot2=12</math>. |
Therefore, our answer is <math>24+12=\boxed{\textbf{(E)} ~36}.</math> | Therefore, our answer is <math>24+12=\boxed{\textbf{(E)} ~36}.</math> | ||
~michaelchang1 | ~michaelchang1 | ||
+ | |||
+ | == Solution 4 (Casework on the Red Chips' Configurations) == | ||
+ | We will consider all possible configurations of the red chips in which rotations matter: | ||
+ | <asy> | ||
+ | /* Made by MRENTHUSIASM */ | ||
+ | unitsize(7mm); | ||
+ | fill((0,2)--(1,2)--(1,3)--(0,3)--cycle, red); | ||
+ | fill((1,1)--(2,1)--(2,2)--(1,2)--cycle, red); | ||
+ | fill((2,0)--(3,0)--(3,1)--(2,1)--cycle, red); | ||
+ | |||
+ | fill((6,2)--(7,2)--(7,3)--(6,3)--cycle, red); | ||
+ | fill((8,2)--(9,2)--(9,3)--(8,3)--cycle, red); | ||
+ | fill((7,1)--(8,1)--(8,2)--(7,2)--cycle, red); | ||
+ | |||
+ | fill((12,2)--(13,2)--(13,3)--(12,3)--cycle, red); | ||
+ | fill((14,2)--(15,2)--(15,3)--(14,3)--cycle, red); | ||
+ | fill((13,0)--(14,0)--(14,1)--(13,1)--cycle, red); | ||
+ | |||
+ | fill((18,2)--(19,2)--(19,3)--(18,3)--cycle, red); | ||
+ | fill((20,1)--(21,1)--(21,2)--(20,2)--cycle, red); | ||
+ | fill((19,0)--(20,0)--(20,1)--(19,1)--cycle, red); | ||
+ | |||
+ | fill((24,1)--(25,1)--(25,2)--(24,2)--cycle, red); | ||
+ | fill((26,1)--(27,1)--(27,2)--(26,2)--cycle, red); | ||
+ | fill((25,2)--(26,2)--(26,3)--(25,3)--cycle, red); | ||
+ | |||
+ | draw((0,0)--(3,0)--(3,3)--(0,3)--cycle, linewidth(1.5)); | ||
+ | draw((0,1)--(3,1), linewidth(1.5)); | ||
+ | draw((0,2)--(3,2), linewidth(1.5)); | ||
+ | draw((1,0)--(1,3), linewidth(1.5)); | ||
+ | draw((2,0)--(2,3), linewidth(1.5)); | ||
+ | |||
+ | draw((6,0)--(9,0)--(9,3)--(6,3)--cycle, linewidth(1.5)); | ||
+ | draw((6,1)--(9,1), linewidth(1.5)); | ||
+ | draw((6,2)--(9,2), linewidth(1.5)); | ||
+ | draw((7,0)--(7,3), linewidth(1.5)); | ||
+ | draw((8,0)--(8,3), linewidth(1.5)); | ||
+ | |||
+ | draw((12,0)--(15,0)--(15,3)--(12,3)--cycle, linewidth(1.5)); | ||
+ | draw((12,1)--(15,1), linewidth(1.5)); | ||
+ | draw((12,2)--(15,2), linewidth(1.5)); | ||
+ | draw((13,0)--(13,3), linewidth(1.5)); | ||
+ | draw((14,0)--(14,3), linewidth(1.5)); | ||
+ | |||
+ | draw((18,0)--(21,0)--(21,3)--(18,3)--cycle, linewidth(1.5)); | ||
+ | draw((18,1)--(21,1), linewidth(1.5)); | ||
+ | draw((18,2)--(21,2), linewidth(1.5)); | ||
+ | draw((19,0)--(19,3), linewidth(1.5)); | ||
+ | draw((20,0)--(20,3), linewidth(1.5)); | ||
+ | |||
+ | draw((24,0)--(27,0)--(27,3)--(24,3)--cycle, linewidth(1.5)); | ||
+ | draw((24,1)--(27,1), linewidth(1.5)); | ||
+ | draw((24,2)--(27,2), linewidth(1.5)); | ||
+ | draw((25,0)--(25,3), linewidth(1.5)); | ||
+ | draw((26,0)--(26,3), linewidth(1.5)); | ||
+ | |||
+ | label("Rotational",(1.5,4.5)); | ||
+ | label("Symmetry",(1.5,3.75)); | ||
+ | label("$2$ configurations",(1.5,-0.75)); | ||
+ | label("$4$ configurations",(7.5,-0.75)); | ||
+ | label("$4$ configurations",(13.5,-0.75)); | ||
+ | label("$4$ configurations",(19.5,-0.75)); | ||
+ | label("$4$ configurations",(25.5,-0.75)); | ||
+ | </asy> | ||
+ | As there are <math>2!=2</math> permutations of blue and green for each configuration, the answer is <math>2\cdot(2+4+4+4+4)=\boxed{\textbf{(E)} ~36}.</math> | ||
+ | |||
+ | ~MRENTHUSIASM (credit given to Flame Kho) | ||
+ | |||
== Video Solution (Easiest) == | == Video Solution (Easiest) == | ||
https://www.youtube.com/watch?v=UPUrYN1YuVA ~ MathEx | https://www.youtube.com/watch?v=UPUrYN1YuVA ~ MathEx |
Latest revision as of 11:18, 25 April 2021
Contents
- 1 Problem
- 2 Solution 1
- 3 Solution 2 (Casework on the Top-Center and Center-Left Chips)
- 4 Solution 3 (Casework and Derangements)
- 5 Solution 4 (Casework on the Red Chips' Configurations)
- 6 Video Solution (Easiest)
- 7 Video Solution by OmegaLearn (Symmetry, Casework, and Reflections/Rotations)
- 8 See Also
Problem
How many ways are there to place indistinguishable red chips, indistinguishable blue chips, and indistinguishable green chips in the squares of a grid so that no two chips of the same color are directly adjacent to each other, either vertically or horizontally?
Solution 1
Call the different colors A,B,C. There are ways to rearrange these colors to these three letters, so must be multiplied after the letters are permuted in the grid. WLOG assume that A is in the center. In this configuration, there are two cases, either all the A's lie on the same diagonal: or all the other two A's are on adjacent corners: In the first case there are two ways to order them since there are two diagonals, and in the second case there are four ways to order them since there are four pairs of adjacent corners.
In each case there is only one way to put the three B's and the three C's as shown in the diagrams. This means that there are ways to arrange A,B, and C in the grid, and there are 6 ways to rearrange the colors. Therefore, there are ways in total, which is .
-happykeeper
Solution 2 (Casework on the Top-Center and Center-Left Chips)
Without the loss of generality, we fix the top-left square with a red chip. We apply casework to its two adjacent chips:
Case (1): The top-center and center-left chips have different colors. There are three sub-cases for Case (1): As there are permutations of the three colors, each sub-case has ways. So, Case (1) has ways in total.
Case (2): The top-center and center-left chips have the same color. There are three sub-cases for Case (2): As there are permutations of the three colors, each sub-case has ways. So, Case (2) has ways in total.
Answer
Together, the answer is
~MRENTHUSIASM
Solution 3 (Casework and Derangements)
Case (1): We have a permutation of R, B, and G as all of the rows. There are ways to rearrange these three colors. After finishing the first row, we move onto the second. Notice how the second row must be a derangement of the first one. By the derangement formula, , so there are two possible permutations of the second row. (Note: You could have also found the number of derangements of PIE). Finally, there are possible permutations for the last row. Thus, there are possibilities.
Case (2): All of the rows have two chips that are the same color and one that is different. There are obviously possible configurations for the first row, for the second, and for the third. .
Therefore, our answer is
~michaelchang1
Solution 4 (Casework on the Red Chips' Configurations)
We will consider all possible configurations of the red chips in which rotations matter: As there are permutations of blue and green for each configuration, the answer is
~MRENTHUSIASM (credit given to Flame Kho)
Video Solution (Easiest)
https://www.youtube.com/watch?v=UPUrYN1YuVA ~ MathEx
Video Solution by OmegaLearn (Symmetry, Casework, and Reflections/Rotations)
https://youtu.be/wKJ9ppI-8Ew ~ pi_is_3.14
See Also
2021 AMC 10A (Problems • Answer Key • Resources) | ||
Preceded by Problem 24 |
Followed by Last Problem | |
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.