Chapter 15.7, Problem 19WE

To determine

To evaluate:

The different $6$ person committees selected from the membership.

Answer to Problem 19WE

  ${}^{16}{C}_3{\times }^{15}{C}_3=254,800$

Explanation of Solution

Given:

A school club has $15$ boys and $16$ girls as members.

Concept used:

Using by the formula of the permutation

  ${}^n{C}_r=\frac{n!}{r!\left(n-r\right)!}$

Calculation:

A school club has $15$ boys and $16$ girls as members

If equal numbers of boys and girls selected

Pick $3$ girls ${=}^{16}{C}_3$

Pick $3$ boys ${=}^{15}{C}_3$

Number of committees

  ${=}^{16}{C}_3{\times }^{15}{C}_3$

  $\begin{array}{l}{}^{16}{C}_3{\times }^{15}{C}_3=\frac{16!}{3!\left(16-3\right)!}\times \frac{15!}{3!\left(15-3\right)!}\\ {}^{16}{C}_3{\times }^{15}{C}_3=\frac{16!}{3!13!}\times \frac{15!}{3!12!}\\ {}^{16}{C}_3{\times }^{15}{C}_3=\frac{16\times 15\times 14\times 13!}{3!13!}\times \frac{16\times 15\times 14\times 13\times 12!}{3!12!}\\ {}^{16}{C}_3{\times }^{15}{C}_3=\frac{16\times 15\times 14}{3\times 2}\times \frac{16\times 15\times 14\times 13}{3\times 2}\\ {}^{16}{C}_3{\times }^{15}{C}_3=16\times 5\times 7\times 16\times 5\times 7\times 13\\ {}^{16}{C}_3{\times }^{15}{C}_3=254,800\end{array}$

Therefore,

The different $6$ person committees selected from the membershipis ${}^{16}{C}_3{\times }^{15}{C}_3=254,800$