Question

in how many ways can a committee of four democrats and five republicans be formed from a group of eleven democrats and nine republicans? a committee of four democrats and five republicans can be formed from a group of eleven democrats and nine republicans in different ways.

Explanation:

Step1: Calculate the number of ways to choose Democrats

The number of ways to choose \(r = 4\) Democrats from \(n=11\) Democrats is given by the combination formula \(C(n,r)=\frac{n!}{r!(n - r)!}\).
So, \(C(11,4)=\frac{11!}{4!(11 - 4)!}=\frac{11\times10\times9\times8}{4\times3\times2\times1}=330\)

Step2: Calculate the number of ways to choose Republicans

The number of ways to choose \(r = 5\) Republicans from \(n = 9\) Republicans is given by the combination formula \(C(n,r)=\frac{n!}{r!(n - r)!}\)
So, \(C(9,5)=\frac{9!}{5!(9 - 5)!}=\frac{9\times8\times7\times6}{4\times3\times2\times1}=126\)

Step3: Use the multiplication principle

Since for each way of choosing Democrats, there are ways of choosing Republicans, the total number of ways to form the committee is the product of the two combinations.
Total ways \(=C(11,4)\times C(9,5)=330\times126 = 41580\)

Answer:

\(41580\)