In how many ways can 12 people be divided into three groups

In how many ways can 12 people be divided into three groups of 4?

Solution

Solution:

should be answered in Combinations because it is question of Combinations not Permutations.

No. of ways 4 can be selected from group of 12 = 12C4

Calculate the combinations for C(n,r) = n! / ( r!(n - r)! ).

C(12,4)=12!/4!(12-4)!=12!/4!8!=12.11.10.9.8.7.6.5.4.3.2.1/(4.3.2.1)(8.7.6.5.4.3.2.1)=495

No. of ways 4 can be selected from remaining 8 = 8C4

No. of ways 4 can be selected from last 4 = 4C4

So total no. of ways = 12C4 * 8C4 * 4C4
= 495* 70*1 = 34650

So 3 groups are formed and order of selection of group is not important. So,

Total no. of ways = 34650 ÷ 3!
= 5775