Coins

In how many ways can 19 coins equal exactly one dollar?

My first thought was let's start with something easy. As I attempted to find a solution,I realized that this problem was not a simple as I thought.

Here was my starting point:

each coin has the following contraints

4 quarters will give me $1.

10 dimes will give me $1.

20 nickels will give me $1.

100 pennies will give me $1.

Trial and Error Method

3 quarters + 2 dimes + 0 nickels + 5 pennies = $1

3 quarters + 2 dimes + 1 nickel + 0 pennies = $1

3 quarters + 1 dime + 0 nickels + 15 pennies = $1***********19 coins

3 quarters + 1 dime + 1 nickel + 10 pennies = $1

3 quarters + 1 dime + 2 nickels + 5 pennies = $1

3 quarters + 1 dime + 3 nickels + 0 pennies = $1

3 quarters + 0 dimes + 2 nickels + 15 pennies = $1

3 quarters + 0 dimes + 3 nickel + 10 pennies = $1

3 quarters + 0 dimes + 4 nickels + 5 pennies = $1

3 quarters + 0 dimes + 5 nickels + 0 pennies = $1

3 quarters + 0 dimes + 0 nickels + 25 pennies = $1

1 way with 3 quarters

2 quarters + 5 dimes + 0 nickel + 0 pennies = $1

2 quarters + 4 dimes + 2 nickels + 0 pennies = $1

2 quarters + 4 dimes + 1 nickels + 5 pennies = $1

2 quarters + 4 dimes + 0 nickels + 10 pennies = $1

2 quarters + 3 dimes + 0 nickel + 20 pennies = $1

2 quarters + 3 dimes + 1 nickels + 15 pennies = $1

2 quarters + 3 dimes + 2 nickels + 10 pennies = $1

2 quarters + 3 dimes + 3 nickels + 5 pennies = $1

2 quarters + 3 dimes + 4 nickels + 0 pennies = $1

I got tired and began to look for another approach!

Could I use Algebraic Methods ?

Let Q = the number of quarters

Let D = the number of dimes

Let N = the number of nickels

Let P = the number of pennies

Q + D + N + P = 19

.25 Q + .10 D + .05 N + .01 P = 1

0 < Q < 4

0 < D < 10

0 < N < 20

0 < P < 100

Solving a system of equations with 4 unknowns would be difficult using algebra.

I resolved that this would be easier on a spread sheet

Click here for spreadsheet solution using Microsoft Excel.

I came up with 6 ways.