First, what coins are available:
p = Penny
n = Nickel
d = Dime
q = Quarter
h = Half Dollar

Some combination of these coins must equal \$1.00, therefore

.01p + .05n + .10d + .25q + .5h = \$1.00
p + n + d + q + h = 19

Questions we should consider:
1. Can we achieve a solution with only one type of coin? No.
2. 100 pennies = \$1
3. 20 nickels = \$1
4. 10 dimes = \$1
5. 4 quarters = \$1
6. 2 half dollars = \$1
7.
What happens if we use pennies?
1. Pennies must be used in multiples of 5, otherwise we cannot reach a whole number
2. Since we cannot use 20 pennies, we can only use 5, 10, or pennies

If we use 5 pennies
1. .05n + .10d + .25q + .5h = \$0.95
2. n + d + q + h = 14
Now consider nickels
1. 14 nickels only equals \$0.70, so we need to reduce the number of nickels
2. If we use 12 nickels
3. .10d + .25q + .5h = \$0.35
4. d + q + h = 2
Now consider the other coins
1. It seems clear from this point that we will need 1 dime and 1 quarter
2. .01(5) + .05(12) + .10(1) + .25(1)= \$1.00

Are there other combinations that can be made using 5 pennies?

Let’s consider nickels:
1. 18 nickels = \$0.90
2. We can only use 1 coin and it must equal \$010, so in this case it must be 1 dime
3. .05(18) + .10(1) = \$1.00
4.
Now let’s look at dimes:
1. 9 dimes = \$0.90
2. We must use 10 more coins and they can’t sum to more than \$0.10
3. 1 nickel and 5 pennies sums to \$0.10, but only requires 6 coins
4. Therefore, the only option is to use 10 pennies
5. .01(10) + .10(9) = \$1.00

Now let’s look at quarters:
1. 3 quarters = \$0.75
2. We must use 16 more coins and they can’t sum to more than \$0.25
3. 2 dimes and 1 nickel sums to \$0.25, but only requires 3 coins
4. 15 pennies and 1 dime has the correct sum and correct number of coins
5. .01(15) + .10(1) + .25(3)= \$1.00
6.
7. 1 quarter = \$0.25
8. We must use 18 more coins and they can’t sum to more than \$0.85
9. 2 dimes and 1 nickel sums to \$0.25, but only requires 3 coins
10. 15 pennies and 1 dime has the correct sum and correct number of coins
11. .01(10) + .05(1) + .10(7) + .25(1)= \$1.00

Now let’s look at half dollars:
1. 1 half dollar = \$0.50
2. We must use 18 more coins and they can’t sum to more than \$0.50
3. 15 pennies and 1 dime sums to \$0.25, but we can’t reach \$1 with only 2 coins
4. 10 pennies, 1 dime, and 1 nickel sums to \$0.25, but does not use enough coins
5. .01(15) + .10(1) + .25(3)= \$1.00
6.
Does adding the equations help?  Coins
Molly McKee     How many ways can 19 coins equal exactly one dollar?
The Problem 