2013 AMC 10A Problem 11

Question

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?

10

12

15

18

25

Answer: A

Concepts:combinations quadratic

Difficulty rating: 1020

Small hint:

Let

n

be the number of council members and use

\binom n2=10

Big hint:

After finding

n

, count

\binom n3

planning committees

Solution:

Let

x

be the number of students. Then the number of ways to pick a two-person committee is

\binom{x}{2} = \dfrac{x(x - 1)}{2}.

We know that this equals

10,

so

x^2 - x = 20

x^2 - x - 20 = 0.

Factoring yields

(x - 5)(x + 4) = 0

x = 5,

since there cannot be a negative number of students.

Then, the number of ways to pick a

3

-person committee is

\binom{5}{3} = \binom{5}{2} = 10.

Thus, A is the correct answer.