Question

a city council consists of six democrats and eight republicans. if a committee of six people is selected, find the probability of selecting three democrats and three republicans.

(type a fraction. simplify your answer.)

Explanation:

Step1: Calculate total number of people

There are 6 Democrats and 8 Republicans, so the total number of people is $6 + 8=14$.

Step2: Calculate the total number of ways to select 6 - person committee

The number of ways to choose 6 people out of 14 is given by the combination formula $C(n,r)=\frac{n!}{r!(n - r)!}$, where $n = 14$ and $r = 6$. So $C(14,6)=\frac{14!}{6!(14 - 6)!}=\frac{14!}{6!8!}=\frac{14\times13\times12\times11\times10\times9}{6\times5\times4\times3\times2\times1}=3003$.

Step3: Calculate the number of ways to select 3 Democrats out of 6

Using the combination formula with $n = 6$ and $r = 3$, we have $C(6,3)=\frac{6!}{3!(6 - 3)!}=\frac{6!}{3!3!}=\frac{6\times5\times4}{3\times2\times1}=20$.

Step4: Calculate the number of ways to select 3 Republicans out of 8

Using the combination formula with $n = 8$ and $r = 3$, we have $C(8,3)=\frac{8!}{3!(8 - 3)!}=\frac{8!}{3!5!}=\frac{8\times7\times6}{3\times2\times1}=56$.

Step5: Calculate the number of favorable cases

The number of ways to select 3 Democrats and 3 Republicans is the product of the number of ways to select 3 Democrats and the number of ways to select 3 Republicans, so $C(6,3)\times C(8,3)=20\times56 = 1120$.

Step6: Calculate the probability

The probability $P$ of selecting 3 Democrats and 3 Republicans is the number of favorable cases divided by the total number of cases. So $P=\frac{C(6,3)\times C(8,3)}{C(14,6)}=\frac{1120}{3003}=\frac{160}{429}$.

Answer:

$\frac{160}{429}$