AMC10 2013 A

AMC10 2013 A · Q11

AMC10 2013 A · Q11. It mainly tests Linear equations, Combinations.

A student council must select a two-person welcoming committee and a three-person planning committee from among its members. There are exactly 10 ways to select a two-person team for the welcoming committee. It is possible for students to serve on both committees. In how many different ways can a three-person planning committee be selected?

学生会需要从其成员中选出一个两人迎接委员会和一个三人规划委员会。有恰好10种方法选出两人迎接委员会。学生可以同时在两个委员会任职。有多少种不同的方法可以选出三人规划委员会？

(A) 10 10

(B) 12 12

(D) 18 18

(E) 25 25

Answer

Correct choice: (A)

正确答案：（A）

Solution

Answer (A): Let $n$ be the number of student council members. Because there are 10 ways of choosing the two-person welcoming committee, it follows that $10=\binom{n}{2}=\frac{1}{2}n(n-1)$, from which $n=5$. The number of ways to select the three-person planning committee is $\binom{5}{3}=10$.

答案（A）：设 $n$ 为学生会成员人数。由于选择两人迎新委员会有 10 种方法，因此有 $10=\binom{n}{2}=\frac{1}{2}n(n-1)$，由此得 $n=5$。选择三人策划委员会的方法数为 $\binom{5}{3}=10$。

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