1. Distributing identical items

Calculate the number of ways to distribute 15 identical dolls and 7 identical
skateboards amongst 6 children, so that each child gets at least one doll.

I was given a solution for this but it doesn't look correct.

1. Give each child one doll, that leaves 9 dolls.
2. Distribute those 9 dolls. This is done with 14 Choose 5. 14 being the 9 dolls + 5 dividers used to divide them up. And then distribute the 5 dividers.
3. Now distribute the skateboards. The answer I have been given for this is 12 Choose 7. I can't see where this comes from.

Is this answer incorrect, because I cant understand how the skateboards are distributed by 12 Choose 7?

2. Re: Distributing identical items

The 12 choose 7 is exactly like the 14 choose 5. You seem to be using the "stars and bars" method. In other words, the different distributions of skateboards look like **|***|*|*|| where this would mean child 1 gets 2 skateboards, child 2 gets 3, child 3 gets 1, child 4 get 1, and child 5 and 6 get 0 skateboards.

Hence to count the number of ways to distribute, you have 12 symbols (7 *'s and 5 bars) and you have to choose what positions to place the 5 bars.. hence 12 choose 5. Alternative you can do 12 choose 7 = 12 choose 5 which would correspond to choosing what positions to put the 7 *'s instead.

If they multiplied correctly then the answer is correct.

Cheers mate.