Question

A committee of seven Congressmen will be selected from a group of six democrats and nine republicans. Find the number of ways of obtaining exactly one democrat

Answer 1

The committee will be selected from = 6 republicans and 1 democrat.

Now, we can select one democrat from a group of six democrats:

6C1

Use combination formula:

$\begin{gathered} 6C1=\frac{6!}{1!(6-5)!} \\ 6C1=6 \end{gathered}$

Then, the number of ways to choose the republicans:

$\begin{gathered} 9C6=\frac{9!}{6!(9-6)!} \\ 9C6=84 \end{gathered}$

Hence, the number of ways to choose the committee consisting of 1 democrat and 6 republicans is 504 ways:

6*84 = 504 ways