Bill has three 50c pieces, two 20c pieces and three 10c pieces. In how many different ways can he make up exactly $1?
Solution
The answer is just three.
Here is how we can work it out:
Look at the 50c coins first. Bill can use one or two of these. If he uses two he has $1 so that is the first way.
Suppose that Bill only uses one 50c piece. Then he has to make up 50c with the other coins. If he uses both 20c coins he has to use a 10c coin. This is the second way.
If Bill uses one 50c coins and one 20c coin he has to use all three 10c coins. This is the third way.
If Bill uses one 50c coin and no 20c coins he doesn’t have enough 10c coins to make up a dollar.
If he doesn’t use a 50c coin, then he doesn’t have enough 20c and 10c pieces to make up a dollar.
Extension:
What coins can Bill have if he can make up a dollar in exactly four ways? Is there more than one possible answer to this question?