Question

a committee of six congressmen will be selected from a group of seven republicans and four democrats. find the number of ways of obtaining a committee with exactly three republicans. there are ways to have exactly three republicans on the committee.

Explanation:

Step1: Calculate the number of ways to choose 3 Republicans

The number of ways to choose \(r\) items from \(n\) items is given by the combination formula \(C(n,r)=\frac{n!}{r!(n - r)!}\).
Here, \(n = 7\) (number of Republicans) and \(r=3\).

$$C(7,3)=\frac{7!}{3!(7 - 3)!}=\frac{7!}{3!4!}=\frac{7\times6\times5\times4!}{3\times2\times1\times4!}=35$$

Step2: Calculate the number of ways to choose 3 Democrats

Since the committee has 6 members and 3 are Republicans, the number of Democrats is \(6-3 = 3\).
Here, \(n = 4\) (number of Democrats) and \(r = 3\).

$$C(4,3)=\frac{4!}{3!(4 - 3)!}=\frac{4!}{3!1!}=\frac{4\times3!}{3!\times1}=4$$

Step3: Use the multiplication principle

The total number of ways to form the committee with exactly 3 Republicans is the product of the number of ways to choose Republicans and the number of ways to choose Democrats.
Let \(N\) be the total number of ways. Then \(N=C(7,3)\times C(4,3)\)
Substitute the values: \(N = 35\times4=140\)

Answer:

\(140\)