Question

A committee of eight congressman will be selected from a group of 11 Democrats and seven Republicans. Find the number of ways of obtaining a committee with exactly 3 Democrats.

Answer 1

`3465`

Explanation:

GIVEN: A committee of eight congressman will be selected from a group of
`11` Democrats and
`7` Republicans.

TO FIND: The number of ways of obtaining a committee with exactly
`3`Democrats.

SOLUTION:

Total number of congressman in committee
`=8`

Total number of republicans available
`=7`

Total number of democrats available
`=11`

As, the committee should have exactly
`3` democrats, there will be
`5` republicans in committee

No. of ways of selecting
`3` democrats out of
`11` democrats
`^1^1C_3=(11!)/((11-3)!3!)`

`=(11!)/(8!3!)`

`=165`

NO. of ways of selecting
`5` republicans out of
`7` republicans
`^7C_5=(7!)/((7-5)!5!)`

`=(7!)/(5!2!)`

`=21`

The number of ways of obtaining a committee with exactly
`3` Democrats.

`=\text{total ways of selecting 3 democrats}*\text{total ways of selecting 5 republicans}`

`=165*21`

`=3465`

Hence there are
`3465` ways to form a committee in which there are exactly
`3` democrats.

Answer 2

Answer: 3,465

Explanation:

Given, Number of Democrats= 11

Number of Republicans = 7

Total persons in the group =18

Number of congressman will be selected from the group = 8

Now , according to the combinations , the number of ways of obtaining a committee with exactly 3 Democrats.= (3 Democrats , 5 Republicans)

`=^(11)C_3*^7C_5\\\\=(11!)/(3!8!)*(7!)/(5!2!)\\\\=(11*10*9*8!)/(6*8!)*(7*6*5!)/(5!*2)\\\\=165*21=3465`

Hence, the number of ways of obtaining a committee with exactly 3 Democrats. = 3,465