We can use either 0, 1, 2, 3, or 4 quarters.

If we use 0 quarters, we can use from 0 up to 10 dimes,
and the rest, if any, in nickels.  That accounts for 11 ways.

If we use 1 quarter, we can use from 0 up to 7 dimes,
and the rest in nickels.  That accounts for 8 ways.

If we use 2 quarters, we can use from 0 up to 5 dimes,
and the rest, if any, in nickels.  That accounts for 6 ways.

If we use 3 quarters, we can use from 0 up to 2 dimes,
and the rest in nickels.  That accounts for 3 ways. 

If we use 4 quarters, that's the whole dollar, so that accounts
for 1 way.

So the total number of ways = 11+8+6+3+1 = 29 

You weren't asked to list them, but here is the 
list of all 29 ways: 

 1.  0 quarters,  0 dimes, and 20 nickels.
 2.  0 quarters,  1 dime,  and 18 nickels.
 3.  0 quarters,  2 dimes, and 16 nickels.
 4.  0 quarters,  3 dimes, and 14 nickels.
 5.  0 quarters,  4 dimes, and 12 nickels.
 6.  0 quarters,  5 dimes, and 10 nickels.
 7.  0 quarters,  6 dimes, and  8 nickels.
 8.  0 quarters,  7 dimes, and  6 nickels.
 9.  0 quarters,  8 dimes, and  4 nickels.
10.  0 quarters,  9 dimes, and  2 nickels.
11.  0 quarters, 10 dimes, and  0 nickels.
12.  1 quarter,   0 dimes, and 15 nickels.
13.  1 quarter,   1 dime,  and 13 nickels.
14.  1 quarter,   2 dimes, and 11 nickels.
15.  1 quarter,   3 dimes, and  9 nickels.
16.  1 quarter,   4 dimes, and  7 nickels.
17.  1 quarter,   5 dimes, and  5 nickels.
18.  1 quarter,   6 dimes, and  3 nickels.
19.  1 quarter,   7 dimes, and  1 nickel.
20.  2 quarters,  0 dimes, and 10 nickels.
21.  2 quarters,  1 dime,  and  8 nickels.
22.  2 quarters,  2 dimes, and  6 nickels.
23.  2 quarters,  3 dimes, and  4 nickels.
24.  2 quarters,  4 dimes, and  2 nickels.
25.  2 quarters,  5 dimes, and  0 nickels.
26.  3 quarters,  0 dimes, and  5 nickels.
27.  3 quarters,  1 dime,  and  3 nickels.
28.  3 quarters,  2 dimes, and  1 nickel.
29.  4 quarters,  0 dimes, and  0 nickels.

Edwin