Need help studying for an actuarial exam:
You have 12 different toys which you want to give to three children so that the oldest chid receives five toys, the middle child four, and the youngest child 3. In how many ways can that be done?
I have a feeling my answer is wrong, but I don't know that to do.
My best guess:
12!/(5! * 7!) + 7!/(4! * 3!) + 1
First you need to pick 5 toys from 12, then 4 from 7, then 3 from 3, to distribute in the way that you describe. That is: $$ {12\choose 5} \times {7\choose 4} \times {3\choose 3} $$ Which is equal to your answer if you change plus signs with multiplication.