Question

A committee of $7$, consisting of $2$ Republicans, $2$ Democrats, and $3$ Independents, is to be chosen from a group of $5$ Republicans, $6$ Democrats, and $4$ Independents. How many committees are possible?

Short answer

The possible number of committees are$600$.

Step-by-step solution

Step 1. Given information.

Here it is given that,

Number of Republicans $=7$

Number of Republicans to be selected $=2$

Number of Democrats $=6$

Number of Democrats to be selected$=2$
Number of Independents$=4$

Number of Independents to be selected$=3$

Step 2. Find the possible number of committees.

Two Republicans can be selected in $\left(\begin{array}{c}5\\ 2\end{array}\right)$ways $=\frac{5!}{2!3!}=\frac{5\times 4\times 3!}{2\times 1\times 3!}=10$

Two Democrats can be selected in$\left(\begin{array}{c}6\\ 2\end{array}\right)$ways$=\frac{6!}{2!4!}=\frac{6\times 5\times 4!}{2\times 1\times 4!}=15$

Three Independents can be selected in role="math" localid="1647788797198" $\left(\begin{array}{c}4\\ 3\end{array}\right)$ways $=\frac{4!}{3!1!}=\frac{4\times 3!}{3!}=4$

Therefore, the the possible number of committees are$=10\times 15\times 4=600$.