In how many ways can 12 balls be distributed to 4 distinguis

In how many ways can 12 balls be distributed to 4 distinguishable (not identical) bags if The balls are distinguishable and each bag gets at least 3 balls?

Solution

There are 4 bags and 12 balls and at least 3 balls go into each bag. Hence all balls go into each bag and exactly 3 balls in each bag.

So problem reduces to dividing the balls into sets of 3 and distributing the sets into bags.

We can choose 3 out of 12 in C(12,3) and put into the first bag

Then 3 out of remaining 9 in C(9,3) and put into second bag

Then 3 out of remaining 6 in C(6,3) and put into third bag

And rest into last bag.

So total number of ways is:

C(12,3)C(9,3)C(6,3)=369600