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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 20:15, 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. Unknown error_msg)

Solution

Desserts must be chosen for $7$ days: Sunday, Monday, Tuesday, Wednesday, Thursday, Friday, Saturday.

There are $3$ 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 $3$ choices for dessert on Thursday. Once dessert for Thursday is selected, there are $3$ choices for dessert on Wednesday, once Wednesday's dessert is selected there are $3$ choices for dessert on Tuesday, etc. Thus, there are $3$ choices for dessert for each of $6$ days, so the total number of possible dessert menus is $3^6$, or $\boxed{\textbf{(A)}\ 729}$.

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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