Question

the senate in a certain state is comprised of 52 republicans, 46 democrats, and 2 independents. how many committees can be formed if each committee must have 3 republicans and 2 democrats?

Explanation:

Step1: Identify the combination formula

The problem involves forming committees with a specific number of Republicans and Democrats, so we use the combination formula \( C(n, k)=\frac{n!}{k!(n - k)!} \), where \( n \) is the total number of items, and \( k \) is the number of items to choose.

Step2: Calculate combinations for Republicans

For Republicans, \( n = 52 \) and \( k = 3 \). So \( C(52, 3)=\frac{52!}{3!(52 - 3)!}=\frac{52\times51\times50}{3\times2\times1}=22100 \).

Step3: Calculate combinations for Democrats

For Democrats, \( n = 46 \) and \( k = 2 \). So \( C(46, 2)=\frac{46!}{2!(46 - 2)!}=\frac{46\times45}{2\times1}=1035 \).

Step4: Multiply the two results

To find the total number of committees, we multiply the number of ways to choose Republicans and Democrats. So \( 22100\times1035 = 22873500 \).

Answer:

\( 22873500 \)