How many different ways can you make 50 cents if you have 1 quarter, 5 dimes, and 10 nickels?

Answer: There are 10 different ways!

Explanation:

You need to find non-negative numbers N, D, and Q so that:

5N + 10D + 25Q = 50

N + 2D + 5Q = 10

Q = 0, 1, 2

If Q = 1, then you have:

N + 2D = 5

Q = 1, D = 2, N = 1

Q = 1, D = 1, N = 3

Q = 1, D = 0, N = 5

N + 2D = 10

This gives the last solutions:

Q = 0, D = 5, N = 0

Q = 0, D = 4, N = 2

Q = 0, D = 3, N = 4

Q = 0, D = 2, N = 6

Q = 0, D = 1, N = 8

Q = 0, D = 0, N = 10