Question

In how many ways can 10 people be divided into three groups of 2, 3, and 5 people respectively

Answer 1

Given:

Total number of people = 10

To find:

The number of ways in which 10 people can be divided into three groups of 2, 3, and 5 people respectively.

Solution:

We know that the number of ways to select r items form n times is:

`^nC_r=(n!)/(r!(n-r)!)`

The number of ways to select 2 people from 10 is
`^(10)C_2`.

The number of remaining people is
`10-2=8`.

The number of ways to select 3 people from 8 is
`^(8)C_3`.

The number of remaining people is
`8-3=5`.

The number of ways to select 5 people from 5 is
`^(5)C_5`.

Now, the total number of ways is:

`Total=^(10)C_2\cdot ^(8)C_3\cdot ^(5)C_5`

`Total=(10!)/(2!(10-2)!)\cdot (8!)/(3!(8-3)!)\cdot (5!)/(5!(5-5)!)`

`Total=(10* 9* 8!)/(2* 1* 8!)\cdot (8* 7* 6* 5!)/(3* 2* 1* 5!)\cdot 1`

`Total=45\cdot 56\cdot 1`

`Total=2520`

Therefore, the total number of ways is 2520 to divide 10 people into three groups of 2, 3, and 5 people respectively.