Question

A club has 30 members is formed from 12 democrats and 18 republicans. How many different committees consisting of 8 democrats
and 8 republicans are possible to be formed?

Answer 1

Given:

Total number of members = 30

Number of democrats = 12

Number of republicans = 18

To find:

Number of different committees consisting of 8 democrats  and 8 republicans are possible to be formed.

Solution:

Total number of ways to selecting 8 democrats  from 12 democrats  is

$^{12}C_8=\dfrac{12!}{8!(12-8)!}$

$^{12}C_8=\dfrac{12\times 11\times 10\times 9\times 8!}{8!4!}$

$^{12}C_8=\dfrac{12\times 11\times 10\times 9}{4\times 3\times 2\times 1}$

$^{12}C_8=495$

Total number of ways to selecting 8 republicans from 18 republicans  is

$^{18}C_8=\dfrac{18!}{8!(18-8)!}$

$^{18}C_8=\dfrac{18\times 17\times 16\times 15\times 14\times 13\times 12\times 11\times 10!}{8!10!}$

$^{18}C_8=\dfrac{18\times 17\times 16\times 15\times 14\times 13\times 12\times 11}{8\times 7\times 6\times 5\times 4\times 3\times 2\times 1}$

$^{18}C_8=43758$

Now, number of different committees consisting of 8 democrats  and 8 republicans are possible to be formed is

$^{12}C_8\times ^{18}C_8=495\times 43758$

$^{12}C_8\times ^{18}C_8=21660210$

Therefore, the total possible ways are 21660210.