Using only quarters, dimes, and nickels, how many different 
combinations of coins can be used to make 75 cents.

If you don't use any quarters you have to have an odd number
of nickels.  That could be any of the odd numbers from 1 through 15
nickels, because you could always make the rest of the 75 cents with
dimes only.  That's 8 ways.

If you use exactly 1 quarter, you have to make the remaining 50 cents
using nickels and dimes and you must have an even number of nickels, from
0 through 10, for you could always make the rest of the 50 cents with
dimes. That's 6 more ways.

If you use 2 quarters, then there are only 3 ways to make the remaining
25 cents with nickels and dimes. That is 2 dimes and a nickel, 3 nickels
and 1 dime, and 5 nickels.  That's 3 more ways.

Finally there is the one way to use 3 quarters.

That's a total of 8+6+3+1 = 18 ways.

Edwin