Difference between revisions of "2012 AMC 10B Problems/Problem 11"
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There are <math>3</math> choices for dessert on Saturday: pie, ice cream, or pudding, as there must be cake on Friday and the same dessert may not be served two days in a row. Likewise, there are <math>3</math> choices for dessert on Thursday. Once dessert for Thursday is selected, there are <math>3</math> choices for dessert on Wednesday, once Wednesday's dessert is selected there are <math>3</math> choices for dessert on Tuesday, etc. Thus, there are <math>3</math> choices for dessert for each of <math>6</math> days, so the total number of possible dessert menus is <math>3^6</math>, or <math>\boxed{\textbf{(A)}\ 729}</math>. | There are <math>3</math> choices for dessert on Saturday: pie, ice cream, or pudding, as there must be cake on Friday and the same dessert may not be served two days in a row. Likewise, there are <math>3</math> choices for dessert on Thursday. Once dessert for Thursday is selected, there are <math>3</math> choices for dessert on Wednesday, once Wednesday's dessert is selected there are <math>3</math> choices for dessert on Tuesday, etc. Thus, there are <math>3</math> choices for dessert for each of <math>6</math> days, so the total number of possible dessert menus is <math>3^6</math>, or <math>\boxed{\textbf{(A)}\ 729}</math>. | ||
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+ | ==See Also== | ||
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+ | {{AMC10 box|year=2012|ab=B|num-b=10|num-a=12}} | ||
{{MAA Notice}} | {{MAA Notice}} |
Revision as of 21:15, 8 February 2014
Problem 11
A dessert chef prepares the dessert for every day of a week starting with Sunday. The dessert each day is either cake, pie, ice cream, or pudding. The same dessert may not be served two days in a row. There must be cake on Friday because of a birthday. How many different dessert menus for the week are possible?
$\textbf{(A)}\ 729\qquad\textbf{(B)}\ 972\qquad\textbf{(C)}\ 1024\qquad\textbf{(D)}\ 2187\qquad\textbf{(E)}\2304$ (Error compiling LaTeX. ! Undefined control sequence.)
Solution
Desserts must be chosen for days: Sunday, Monday, Tuesday, Wednesday, Thursday, Friday, Saturday.
There are choices for dessert on Saturday: pie, ice cream, or pudding, as there must be cake on Friday and the same dessert may not be served two days in a row. Likewise, there are choices for dessert on Thursday. Once dessert for Thursday is selected, there are choices for dessert on Wednesday, once Wednesday's dessert is selected there are choices for dessert on Tuesday, etc. Thus, there are choices for dessert for each of days, so the total number of possible dessert menus is , or .
See Also
2012 AMC 10B (Problems • Answer Key • Resources) | ||
Preceded by Problem 10 |
Followed by Problem 12 | |
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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