Difference between revisions of "2022 AMC 10A Problems/Problem 9"
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==Problem== | ==Problem== | ||
| − | 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> | + | A rectangle is partitioned into <math>5</math> 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? |
| − | size(5.5cm); | + | |
| − | draw((0,0)--(0,2)--(2,2)--(2,0)--cycle); | + | <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> |
| − | 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> | ||
<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> | ||
Revision as of 00:15, 12 November 2022
Problem
A rectangle is partitioned into 
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/c/5/3/c53adae322a3fcf0e0d1d0d2aa43ce0ad69527c9.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 | ||
| 2022 AMC 12A (Problems • Answer Key • Resources) | |
| Preceded by Problem 6 |
Followed by Problem 8 |
| 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 12 Problems and Solutions | |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 
