Question

the senate in a certain state is comprised of 51 republicans, 46 democrats, and 3 independents. how many committees can be formed if each committee must have 3 republicans and 2 democrats? committees can be formed.

Explanation:

Step1: Calculate the number of ways to choose 3 Republicans

The formula for combinations is \(C(n,r)=\frac{n!}{r!(n - r)!}\), where \(n = 51\) (number of Republicans) and \(r=3\).

$$C(51,3)=\frac{51!}{3!(51 - 3)!}=\frac{51\times50\times49}{3\times2\times1}=20825$$

Step2: Calculate the number of ways to choose 2 Democrats

Here \(n = 46\) (number of Democrats) and \(r = 2\)

$$C(46,2)=\frac{46!}{2!(46- 2)!}=\frac{46\times45}{2\times1}=1035$$

Step3: Use the multiplication principle

The total number of committees is the product of the number of ways to choose Republicans and Democrats.

$$N = C(51,3)\times C(46,2)=20825\times1035 = 21553875$$

Answer:

\(21553875\)