how many different combinations can you make using pennies, nickels, dimes, quarters, and the half dollar that equal 60 cents?

No idea. Just start listing them, fewest coins first. Then start breaking down into smaller values

50 10
50 5 5
50 5 1 1 1 1 1
50 1 1 1 1 1 1 1 1 1 1
25 25 10
...

To find the number of different combinations of coins that equal 60 cents, we can use a systematic approach. Since you have the following coins: pennies, nickels, dimes, quarters, and half dollars, let's analyze each coin's value.

1. Pennies: Pennies have a value of 1 cent each. To reach 60 cents using only pennies, you would require 60 pennies. However, since we have other coins available, we won't consider using only pennies.

2. Nickels: Nickels have a value of 5 cents each. To reach 60 cents, using only nickels, you would need 12 nickels since 5 * 12 = 60. However, we have other coins available, so we won't consider using only nickels either.

3. Dimes: Dimes have a value of 10 cents each. To reach 60 cents, using only dimes, you would need 6 dimes since 10 * 6 = 60. Again, we have other coins available, so using only dimes won't give us different combinations.

4. Quarters: Quarters have a value of 25 cents each. To reach 60 cents, using only quarters, you would require 2 quarters since 25 * 2 = 50. However, using only quarters won't give us different combinations.

5. Half dollars: Half dollars have a value of 50 cents each. Using a single half dollar would already exceed the target of 60 cents, so we won't consider using half dollars for this combination.

Now, let's consider various combinations of the available coins:

a) Pennies and nickels: We can use a couple of pennies with various numbers of nickels. By systematically increasing the number of nickels, we find that the possible combinations are:
- 2 pennies + 10 nickels = 60 cents
- 7 pennies + 8 nickels = 60 cents
- 12 pennies + 6 nickels = 60 cents
...and so on.

b) Pennies and dimes: Similar to the previous case, we can combine pennies and dimes. By systematically increasing the number of dimes, we find the following combinations:
- 1 penny + 5 dimes = 60 cents
- 6 pennies + 3 dimes = 60 cents
- 11 pennies + 1 dime = 60 cents

c) Pennies, nickels, and dimes: Here, we can combine all three coins. By systematically increasing the number of each coin, we can find additional combinations. For instance:
- 2 pennies + 5 nickels + 3 dimes = 60 cents
- 7 pennies + 3 nickels + 1 dime = 60 cents

Using this systematic approach, you can continue to explore various combinations until you have considered all possibilities.