Question

the senate in a certain state is comprised of 52 republicans, 47 democrats, and 1 independent. 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 Republicans

The number of ways to choose \(3\) Republicans out of \(52\) is given by the combination formula \(C(n,r)=\frac{n!}{r!(n - r)!}\), where \(n = 52\) and \(r=3\).

$$C(52,3)=\frac{52!}{3!(52 - 3)!}=\frac{52\times51\times50}{3\times2\times1}=22100$$

Step2: Calculate the number of ways to choose Democrats

The number of ways to choose \(2\) Democrats out of \(47\) is given by the combination formula \(C(n,r)=\frac{n!}{r!(n - r)!}\), where \(n = 47\) and \(r = 2\).

$$C(47,2)=\frac{47!}{2!(47- 2)!}=\frac{47\times46}{2\times1}=1081$$

Step3: Calculate the total number of committees

By the multiplication principle, the total number of committees is the product of the number of ways to choose Republicans and the number of ways to choose Democrats.

$$N = C(52,3)\times C(47,2)=22100\times1081 = 23880100$$

Answer:

\(23880100\) committees can be formed.