AMC10 2003 A

AMC10 2003 A · Q21

AMC10 2003 A · Q21. It mainly tests Basic counting (rules of product/sum), Combinations.

Pat is to select six cookies from a tray containing only chocolate chip, oatmeal, and peanut butter cookies. There are at least six of each of these three kinds of cookies on the tray. How many different assortments of six cookies can be selected?

Pat 要从一个盘子里选择六块饼干，盘子里只有巧克力碎片、燕麦和花生酱饼干，每种至少有六块。有多少种不同的六块饼干组合可以选择？

(A) 22 22

(B) 25 25

(D) 28 28

(E) 729 729

Answer

Correct choice: (D)

正确答案：（D）

Solution

(D) The numbers of the three types of cookies must have a sum of six. Possible sets of whole numbers whose sum is six are $0,0,6;\ 0,1,5;\ 0,2,4;\ 0,3,3;\ 1,1,4;\ 1,2,3;\ \text{and}\ 2,2,2.$ Every ordering of each of these sets determines a different assortment of cookies. There are 3 orders for each of the sets $0,0,6;\ 0,3,3;\ \text{and}\ 1,1,4.$ There are 6 orders for each of the sets $0,1,5;\ 0,2,4;\ \text{and}\ 1,2,3.$ There is only one order for $2,2,2$. Therefore the total number of assortments of six cookies is $3\cdot 3+3\cdot 6+1=28$.

（D）三种类型饼干的数量之和必须为 6。所有和为 6 的非负整数集合可以是 $0,0,6;\ 0,1,5;\ 0,2,4;\ 0,3,3;\ 1,1,4;\ 1,2,3;\ \text{以及}\ 2,2,2.$ 上述每个集合的不同排列对应一种不同的饼干组合。以下每个集合各有 3 种排列： $0,0,6;\ 0,3,3;\ \text{以及}\ 1,1,4.$ 以下每个集合各有 6 种排列： $0,1,5;\ 0,2,4;\ \text{以及}\ 1,2,3.$ $2,2,2$ 只有一种排列。因此，6 块饼干的组合总数为 $3\cdot 3+3\cdot 6+1=28$。

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