# Difference between revisions of "2012 AMC 10B Problems/Problem 11"

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== Solution == | == Solution == | ||

− | <math>\boxed{\textbf{(A)}\ 729}</math> | + | Desserts must be chosen for <math>7</math> days: Sunday, Monday, Tuesday, Wednesday, Thursday, Friday, Saturday. |

+ | |||

+ | 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>. |

## Revision as of 19:16, 25 February 2012

## 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 .