How many ways are there to make 33 cents using any combination of pennies, nickels, and dimes?

1 Answer



The only way I see to do this is to chart it out, but there are a couple of short cuts to help along the way:

1 dime = 2 nickels = 5 pennies

I'll do this using D for dimes, N for nickels and P for pennies:

3D 0N 3P

2D 2N 3P
2D 1N 8P
2D 0N 13P

1D 4N 3P
1D 3N 8P
1D 2N 13P
1D 1N 18P
1D 0N 23P

now let's stop here for a second and see that we have a pattern of 1, 3, then 5 ways. I'll make a prediction that it'll be 7 ways with 0D:

0D 6N 3P

and so on - we'll work our way down from 6N to 0N, which is 7 ways.

Total number of ways is: