Question

In how many ways can 12 one-dollar coins be shared between Josh, Frank, and Suzy if each of them receives at least 3 coins?

Answer 1

Since 9 of the coins are accounted for (each must receive at least 3) let's see how we can divide the remaining 3 coins.

Josh Frank Suzy
3 0 0 Josh gets all of them
0 3 0 Frank gets all of them
0 0 3 Suzy gets all of them
1 1 1 They share them
2 1 0 2 way to do it if Josh gets 2 coins
2 0 1 (this is the 2nd way if Josh gets two)

So there must be two ways to do it if Frank gets 2 coins and two more ways to do it if Suzy gets two coins.

Recap: 3 way if one person gets all 3 coins and the others get 0
1 way if they all get a single coin
6 ways if one person gets 2 and another gets the remaining one.

Total = 10 ways.