Question

in how many ways can a committee of five democrats and five republicans be formed from a group of eight democrats and eight republicans? a committee of five democrats and five republicans can be formed from a group of eight democrats and eight republicans in \boxed{} different ways.

Explanation:

Step1: Calculate the number of ways to choose 5 Democrats from 8 Democrats

The formula for combinations is \(C(n,r)=\frac{n!}{r!(n - r)!}\), where \(n = 8\) (total number of Democrats) and \(r=5\) (number of Democrats to be chosen).

$$C(8,5)=\frac{8!}{5!(8 - 5)!}=\frac{8!}{5!3!}=\frac{8\times7\times6\times5!}{5!\times3\times2\times1}=56$$

Step2: Calculate the number of ways to choose 5 Republicans from 8 Republicans

Using the combination formula with \(n = 8\) (total number of Republicans) and \(r = 5\) (number of Republicans to be chosen)

$$C(8,5)=\frac{8!}{5!(8 - 5)!}=\frac{8!}{5!3!}=\frac{8\times7\times6\times5!}{5!\times3\times2\times1}=56$$

Step3: Use the multiplication principle

Since the choices of Democrats and Republicans are independent events, the total number of ways to form the committee is the product of the number of ways to choose Democrats and the number of ways to choose Republicans.

$$N = C(8,5)\times C(8,5)=56\times56 = 3136$$

Answer:

3136