Friend hit me with this problem today.

There are three children with 20 fruit (10 peaches, 4 apples, and 6 pineapples).

How many ways can the fruit be passed out so a child has at least one of any kind of fruit?

My approach (which is wrong) is:

nCr(9,7)*nCr(6,4)*nCr(8,6) + nCr(12,10)*nCr(3,1)*nCr(8,6) + nCr(12,10)*nCr(6,4)*nCr(5,3)

(AKA, front load every kid 1 peach each, plus 1 apple each, plus 1 pineapple each)

This will count the number of ways each child is guaranteed one fruit. However it is over counting because when I front load 3 of any kind of fruit to a kid it is also re-counted in the other group when i front load another 3 fruit. Any advice how I can take out the over count?

PS: How do I use the awesome text so I don't have to use nCr?