2013 AMC 10A Problems/Problem 11

Problem

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?

$\textbf{(A)}\ 10\qquad\textbf{(B)}\ 12\qquad\textbf{(C)}\ 15\qquad\textbf{(D)}\ 18\qquad\textbf{(E)}\ 25$

Solution

Let the number of students on the council be $x$ . We know that there are $\dbinom{x}{2}$ ways to choose a two person team. This gives that $x(x-1) = 20$ , which has a positive integer solution of $5$ .

If there are $5$ people on the welcoming committee, then there are $\dbinom{5}{3} = 10$ ways to choose a three-person committee, $\textbf{(A)}$ .

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2013 AMC 10A Problems/Problem 11

Problem

Solution