AMC12 2012 B

AMC12 2012 B · Q8

AMC12 2012 B · Q8. It mainly tests Basic counting (rules of product/sum).

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?

一位甜点师从星期日开始为一周的每天准备甜点。每天的甜点要么是蛋糕、派、冰淇淋或布丁。相同的甜点不能连续两天供应。由于生日，星期五必须供应蛋糕。一周可能有多少种不同的甜点菜单？

(A) 729 729

(B) 972 972

(D) 2187 2187

(E) 2304 2304

Answer

Correct choice: (A)

正确答案：（A）

Solution

We can count the number of possible foods for each day and then multiply to enumerate the number of combinations. On Friday, we have one possibility: cake. On Saturday, we have three possibilities: pie, ice cream, or pudding. This is the end of the week. On Thursday, we have three possibilities: pie, ice cream, or pudding. We can't have cake because we have to have cake the following day, which is the Friday with the birthday party. On Wednesday, we have three possibilities: cake, plus the two things that were not eaten on Thursday. Similarly, on Tuesday, we have three possibilities: the three things that were not eaten on Wednesday. Likewise on Monday: three possibilities, the three things that were not eaten on Tuesday. On Sunday, it is tempting to think there are four possibilities, but remember that cake must be served on Friday. This serves to limit the number of foods we can eat on Sunday, with the result being that there are three possibilities: The three things that were not eaten on Monday. So the number of menus is $3 \cdot 3 \cdot 3 \cdot 3 \cdot 3 \cdot 1 \cdot 3 = 729.$ The answer is $\boxed{A}$.

我们可以分别计算每天可能的甜点数目，然后相乘得到总组合数。星期五只有一种可能：蛋糕。星期六有三种可能：派、冰淇淋或布丁。这是一周的最后一天。星期四有三种可能：派、冰淇淋或布丁。不能是蛋糕，因为第二天（星期五）必须是蛋糕。星期三有三种可能：蛋糕，以及星期四没有吃的另外两种。同理，星期二有三种可能：星期三没有吃的三种。星期一同样：三种可能，即星期二没有吃的三种。星期日看似有四种可能，但要记得星期五必须供应蛋糕。这会限制星期日可选的甜点，因此星期日也只有三种可能：星期一没有吃的三种。所以菜单总数为 $3 \cdot 3 \cdot 3 \cdot 3 \cdot 3 \cdot 1 \cdot 3 = 729.$ 答案是 $\boxed{A}$。

Topics

CP.001 Basic counting (rules of product/sum)