SOLUTION: What are the ways to generate 1 dollar using pennies, nickels, dimes, quarters, and half dollars? What is a mathematical method to find these?

Question 560990: What are the ways to generate 1 dollar using pennies, nickels, dimes, quarters, and half dollars? What is a mathematical method to find these?
Answer by richard1234(7193) (Show Source): You can put this solution on YOUR website!
We can let p, n, d, q, and h be the number of pennies, ..., half dollars respectively. Hence we have

First, we can remove p because the number of pennies will be fixed once we decide how many of the other coins to use. Note that p must be a multiple of 5 because every other term in the equation is divisible by 5.

We have 5n + 10d + 25q + 50h = 5k, where k ranges from 0 to 20, inclusive. We can divide everything by 5 to obtain n + 2d + 5q + 10h = k. Here, it is easier to count the number of ways to form a valid equation. I believe the number of solutions for each k is some recursive function but I don't remember too well.

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You can probably generate a recursive formula for the number of ways to create n cents. Note that the number of ways to make 101 cents, ..., 104 cents is the same as the number of ways to make $1, because only the number of pennies increases. Once you get to 105 cents, it becomes different.

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