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?

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.