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?

$10$

$12$

$15$

$18$

$25$

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.

2013 AMC 10A Problem 11

Solution:

Problem 11 in Other Years