Difference between revisions of "2022 AMC 10A Problems/Problem 9"
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<math>\textbf{(A) }120\qquad\textbf{(B) }270\qquad\textbf{(C) }360\qquad\textbf{(D) }540\qquad\textbf{(E) }720</math> | <math>\textbf{(A) }120\qquad\textbf{(B) }270\qquad\textbf{(C) }360\qquad\textbf{(D) }540\qquad\textbf{(E) }720</math> | ||
| − | ==Solution | + | |
| + | ==Solution== | ||
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>. | 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>. | ||
~Txu | ~Txu | ||
| + | |||
| + | == See Also == | ||
| + | |||
| + | {{AMC10 box|year=2022|ab=A|num-b=8|num-a=10}} | ||
| + | {{MAA Notice}} | ||
Revision as of 21:43, 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]](http://latex.artofproblemsolving.com/0/d/2/0d2429901955e7311077329c6e6b81a1939b686a.png)
Solution
The top left rectangle can be 
possible colors. Then the bottom left region can only be 
possible colors, and the bottom middle can only be 
colors since it is next to the top left and bottom left. Similarly, we have choices for the top right and choices for the bottom right, which gives us a total of 
.
~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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