Nickels Cent

How many ways can you make 78 cents with nickels, dimes, and/or quarters?

ok my teacher asked this in class and well… idk the answer help!

Start by using as many quarters as possible.
3Q + 3P

Now switch out a quarter and replace everything else with as many dimes as possible, then nickels and pennies:
2Q + 2D + 1N + 3P

Alternate by replacing a nickel with 5 pennies, then replacing a dime with two nickels:
2Q + 2D + 8P
2Q + 1D + 3N + 3P
2Q + 1D + 2N + 8P
2Q + 1D + 1N + 13P
2Q + 1D + 18P
2Q + 5N + 3P
2Q + 4N + 8P
2Q + 3N + 13P
2Q + 2N + 18P
2Q + 1N + 23P
2Q + 28P

Now remove another quarter, replace with as many dimes as possible and start the process again:
1Q + 5D + 3P
1Q + 4D + 2N + 3P
1Q + 4D + 1N + 8P
1Q + 4D + 13P
1Q + 3D + 4N + 3P
1Q + 3D + 3N + 8P
1Q + 3D + 2N + 13P
1Q + 3D + 1N + 18P
1Q + 3D + 23P
1Q + 2D + 6N + 3P
1Q + 2D + 5N + 8P
1Q + 2D + 4N + 13P
1Q + 2D + 3N + 18P
1Q + 2D + 2N + 23P
1Q + 2D + 1N + 28P
1Q + 2D + 33P
1Q + 1D + 8N + 3P
1Q + 1D + 7N + 8P
1Q + 1D + 6N + 13P
1Q + 1D + 5N + 18P
1Q + 1D + 4N + 23P
1Q + 1D + 3N + 28P
1Q + 1D + 2N + 33P
1Q + 1D + 1N + 38P
1Q + 1D + 43P
1Q + 10N + 3P
1Q + 9N + 8P
1Q + 8N + 13P
1Q + 7N + 18P
1Q + 6N + 23P
1Q + 5N + 28P
1Q + 4N + 33P
1Q + 3N + 38P
1Q + 2N + 43P
1Q + 1N + 48P
1Q + 53P

Okay, then finally get rid of all the quarters and only do dimes and nickels and pennies:
7D + 1N + 3P
7D + 8P
6D + 3N + 3P
6D + 2N + 8P
6D + 1N + 13P
6D + 18P
5D + 5N + 3P
5D + 4N + 8P
5D + 3N + 13P
5D + 2N + 18P
5D + 1N + 23P
5D + 28P
4D + 7N + 3P
4D + 6N + 8P
4D + 5N + 13P
4D + 4N + 18P
4D + 3N + 23P
4D + 2N + 28P
4D + 1N + 33P
4D + 38P
3D + 9N + 3P
3D + 8N + 8P
3D + 7N + 13P
3D + 6N + 18P
3D + 5N + 23P
3D + 4N + 28P
3D + 3N + 33P
3D + 2N + 38P
3D + 1N + 43P
3D + 48P
2D + 11N + 3P
2D + 10N + 8P
2D + 9N + 13P
2D + 8N + 18P
2D + 7N + 23P
2D + 6N + 28P
2D + 5N + 33P
2D + 4N + 38P
2D + 3N + 43P
2D + 2N + 48P
2D + 1N + 53P
2D + 58P
1D + 13N + 3P
1D + 12N + 8P
1D + 11N + 13P
1D + 10N + 18P
1D + 9N + 23P
1D + 8N + 28P
1D + 7N + 33P
1D + 6N + 38P
1D + 5N + 43P
1D + 4N + 48P
1D + 3N + 53P
1D + 2N + 58P
1D + 1N + 63P
1D + 68P
15N + 3P
14N + 8P
13N + 13P
12N + 18P
11N + 23P
10N + 28P
9N + 33P
8N + 38P
7N + 43P
6N + 48P
5N + 53P
4N + 58P
3N + 63P
2N + 68P
1N + 73P
78P

Whew! If you add them all up, I count 121 ways!

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