Question

a city council consists of eight democrats and seven republicans. if a committee of four people is selected, find the probability of selecting two democrats and two republicans. (type a fraction. simplify your answer.)

Explanation:

Step1: Calculate total number of council - members

The total number of council - members is \(8 + 7=15\). The number of ways to choose 4 people out of 15 is given by the combination formula \(C(n,r)=\frac{n!}{r!(n - r)!}\), where \(n = 15\) and \(r = 4\). So, \(C(15,4)=\frac{15!}{4!(15 - 4)!}=\frac{15!}{4!×11!}=\frac{15\times14\times13\times12}{4\times3\times2\times1}=1365\).

Step2: Calculate number of ways to choose 2 Democrats out of 8

Using the combination formula with \(n = 8\) and \(r = 2\), we have \(C(8,2)=\frac{8!}{2!(8 - 2)!}=\frac{8!}{2!×6!}=\frac{8\times7}{2\times1}=28\).

Step3: Calculate number of ways to choose 2 Republicans out of 7

Using the combination formula with \(n = 7\) and \(r = 2\), we have \(C(7,2)=\frac{7!}{2!(7 - 2)!}=\frac{7!}{2!×5!}=\frac{7\times6}{2\times1}=21\).

Step4: Calculate number of favorable cases

The number of ways to choose 2 Democrats and 2 Republicans is the product of the number of ways to choose 2 Democrats and the number of ways to choose 2 Republicans. So, the number of favorable cases is \(C(8,2)\times C(7,2)=28\times21 = 588\).

Step5: Calculate the probability

The probability \(P\) of choosing 2 Democrats and 2 Republicans is the number of favorable cases divided by the total number of cases. So, \(P=\frac{588}{1365}=\frac{84}{195}=\frac{28}{65}\).

Answer:

\(\frac{28}{65}\)