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=(12!)/(8!(12-8)!)`

`^(12)C_8=(12* 11* 10* 9* 8!)/(8!4!)`

`^(12)C_8=(12* 11* 10* 9)/(4* 3* 2* 1)`

`^(12)C_8=495`

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

`^(18)C_8=(18!)/(8!(18-8)!)`

`^(18)C_8=(18* 17* 16* 15* 14* 13* 12* 11* 10!)/(8!10!)`

`^(18)C_8=(18* 17* 16* 15* 14* 13* 12* 11)/(8* 7* 6* 5* 4* 3* 2* 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* ^(18)C_8=495* 43758`

`^(12)C_8* ^(18)C_8=21660210`

Therefore, the total possible ways are 21660210.