We just need to think about the quarters and dimes for the rest can
always be made up with nickels.  There are three cases, using
0,1, and 2 quarters:

1.  If you don't use any quarters, then you can use 0 dimes
up through 5 dimes, (and the rest, if any, in nickels).  
That's 6 ways.

2. If you use 1 quarter, you can use 0 dimes up through 2 dimes,
(and the rest in nickels).  That's 3 more ways.

3. If you use 2 quarters, that's the whole 50 cents, so that's 1 more way.

So that's 6+3+1 or 10 ways. 

 1. 0 quarters, 0 dimes, 10 nickels.
 2. 0 quarters, 1 dime,   8 nickels.
 3. 0 quarters, 2 dimes,  6 nickels.
 4. 0 quarters, 3 dimes,  4 nickels.
 5. 0 quarters, 4 dimes,  2 nickels.
 6. 0 quarters, 5 dimes,  0 nickels.
 7. 1 quarter,  0 dimes,  5 nickels.
 8. 1 quarter,  1 dime,   3 nickels.
 9. 1 quarter,  2 dimes,  1 nickel.
10. 2 quarters, 0 dimes,  0 nickels.

Edwin