You only need to look at the number of dimes and nickels used,
for you can always use enough pennies to finish out the 25 cents.

There are three cases, using 0, 1, or 2 dimes.

1. If you don't use any dimes, you can use from 0 though 5 nickels,
That's 6 ways.

2. If you use one dime, you can use from 0 though 3 nickels,
That's 4 ways.

3. If you use two dimes, you can use 0 or 1 nickel
That's 2 ways.

So altogether there are 6+4+2 or 12 ways.  Here they all are:

1.  0 dimes, 0 nickels, and 25 pennies.
2.  0 dimes, 1 nickel,  and 20 pennies.
3.  0 dimes, 2 nickels, and 15 pennies.
4.  0 dimes, 3 nickels, and 10 pennies.
5.  0 dimes, 4 nickels, and 5 pennies.
6.  0 dimes, 5 nickels, and 0 pennies.
7.  1 dime,  0 nickels, and 15 pennies.
8.  1 dime,  1 nickel,  and 10 pennies.
9.  1 dime,  2 nickels, and 5 pennies.
10. 1 dime,  3 nickels, and 0 pennies.
11. 2 dimes, 0 nickels, and 5 pennies.
12. 2 dimes, 1 nickel,  and 0 pennies.


Edwin