Question

A city council consists of seven Democrats and five Republicans. If a committee of four people is selected, find the probability
selecting two Democrats and two Republicans.

Answer 1

Answer:

The probability  of selecting two Democrats and two Republicans is 0.4242.

Step-by-step explanation:

The information provided is as follows:

• A city council consists of seven Democrats and five Republicans.
• A committee of four people is selected.

In mathematics, the procedure to select k items from n distinct items, without replacement, is known as combinations.

The formula to compute the combinations of k items from n is given by the formula:

${n\choose k}=\frac{n!}{k!\times (n-k)!}$

Compute the number of ways to select four people as follows:

${12\choose 4}=\frac{12!}{4!\times (12-4)!}=495$

Compute the number of ways to selected two Democrats as follows:

${7\choose 2}=\frac{7!}{2!\times (7-2)!}=21$

Compute the number of ways to selected two Republicans as follows:

${5\choose 2}=\frac{5!}{2!\times (5-2)!}=10$

Then the probability  of selecting two Democrats and two Republicans as follows:

$P(\text{2 Democrats and 2 Republicans})=\frac{21\times 10}{495}=0.4242$

Thus, the probability  of selecting two Democrats and two Republicans is 0.4242.