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

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

80x^8−68x^6+16^5−20x^4−4x^3+8x^2

hope i'm right lol