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Difference between revisions of "2022 AMC 10A Problems/Problem 9"
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==Solution==
 
==Solution==
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The top left rectangle can be <math>5</math> possible colors. Then the bottom left region can only be <math>4</math> possible colors, and the bottom middle can only be <math>3</math> colors since it is next to the top left and bottom left. Similarly, we have <math>3</math> choices for the top right and <math>3</math> choices for the bottom right, which gives us a total of <math>5\cdot4\cdot3\cdot3\cdot3=\boxed{540}</math>.
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The top left rectangle can be <math>5</math> possible colors. Then the bottom left region can only be <math>4</math> possible colors, and the bottom middle can only be <math>3</math> colors since it is next to the top left and bottom left. Similarly, we have <math>3</math> choices for the top right and <math>3</math> choices for the bottom right, which gives us a total of <math>5\cdot4\cdot3\cdot3\cdot3=\boxed{\textbf{(D) }540}</math>.
 
~Txu
 
~Txu
  

Revision as of 21:49, 11 November 2022

A rectangle is partitioned into 5 regions as shown. Each region is to be painted a solid color - red, orange, yellow, blue, or green - so that regions that touch are painted different colors, and colors can be used more than once. How many different colorings are possible? [asy] size(5.5cm); draw((0,0)--(0,2)--(2,2)--(2,0)--cycle); draw((2,0)--(8,0)--(8,2)--(2,2)--cycle); draw((8,0)--(12,0)--(12,2)--(8,2)--cycle); draw((0,2)--(6,2)--(6,4)--(0,4)--cycle); draw((6,2)--(12,2)--(12,4)--(6,4)--cycle); [/asy]

$\textbf{(A) }120\qquad\textbf{(B) }270\qquad\textbf{(C) }360\qquad\textbf{(D) }540\qquad\textbf{(E) }720$

Solution

The top left rectangle can be $5$ possible colors. Then the bottom left region can only be $4$ possible colors, and the bottom middle can only be $3$ colors since it is next to the top left and bottom left. Similarly, we have $3$ choices for the top right and $3$ choices for the bottom right, which gives us a total of $5\cdot4\cdot3\cdot3\cdot3=\boxed{\textbf{(D) }540}$. ~Txu

See Also

2022 AMC 10A (Problems • Answer Key • Resources)
Preceded by
Problem 8 		Followed by
Problem 10
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

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