2012 AMC 10B Problem 11

Problem 11 of 25IntermediateCounting & Probability

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?

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Solution

We know that there must be cake served on Friday, and as such, on Saturday, we cannot have cake. Therefore, we have

3

choices for Saturday’s menu. Furthermore, for each of the

5

previous days, we could go backwards and have

3

choices on each day, making the total number of choices

3^5\cdot 3 = 729.

Thus, the correct answer is A .

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