AMC10 2012 B

AMC10 2012 B · Q11

AMC10 2012 B · Q11. It mainly tests Basic counting (rules of product/sum), Recursion & DP style counting (basic).

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?

一位甜点厨师为一周从星期日开始的每一天准备甜点。每一天的甜点是蛋糕、派、冰淇淋或布丁之一。不能连续两天提供相同的甜点。因为生日的原因，星期五必须有蛋糕。一周的甜点菜单有多少种不同的可能？

(A) 729 729

(B) 972 972

(D) 2187 2187

(E) 2304 2304

Answer

Correct choice: (A)

正确答案：（A）

Solution

Answer (A): There are 3 choices for Saturday (anything except cake) and for the same reason 3 choices for Thursday. Similarly there are 3 choices for Wednesday, Tuesday, Monday, and Sunday (anything except what was to be served the following day). Therefore there are $3^6=729$ possible dessert menus.

答案（A）：星期六有 3 种选择（除蛋糕外的任何甜点），同理星期四也有 3 种选择。类似地，星期三、星期二、星期一和星期日也各有 3 种选择（除去第二天将要供应的那种甜点）。因此共有 $3^6=729$ 种可能的甜点菜单。

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