If 9 red balloons 6 blue balloons and 12 yellow balloons are

If 9 red balloons, 6 blue balloons and 12 yellow balloons are to be distributed between 4 children, how many distributions are possible if every child must receive at least one balloon of each color?

Solution

We first give each child one ballon of each color.

This leaves us to play 5 red, 2 blue, 8 yellow ballons to be distributed in any way to each of the 4 children. There are 15 balloons to 4 children.

Using balls and urns formula, for n balls and r urns, there are

#ways = (n + r - 1)! / (n! (r-1)!) = (15+4-1)!/(15!(4-1)!)

#ways = 816

As we can still permute the 15 balloons in 15!/(5!2!8!) ways by permutation of like objects, then the total number of ways is

#ways(total) = 816*15!/(5!2!8!) = 135135 [ANSWER]