how many ways are there to make 40 cent using dimes, nikels, and pennies

To calculate the number of ways to make 40 cents using dimes, nickels, and pennies, we can use a combination of systematic counting and basic mathematical principles.

First, let's determine the denominations we can use to make 40 cents:

1. Dime: Worth 10 cents
2. Nickel: Worth 5 cents
3. Penny: Worth 1 cent

To find all the possible combinations, we can start by considering the number of dimes we can use.

Case 1: No dimes are used
In this case, we can only use nickels and pennies. Since we need to make 40 cents, we can start by using nickels and then adding pennies to the remaining amount. The possible combinations are as follows:

- 8 nickels: No pennies needed.
- 6 nickels: Add 1 penny to the remaining 5 cents.
- 4 nickels: Add 2 pennies to the remaining 10 cents.
- 2 nickels: Add 3 pennies to the remaining 15 cents.
- 0 nickels: Add 4 pennies to the remaining 20 cents.

Case 2: One dime is used
In this case, the remaining amount to be made is 30 cents. Similar to Case 1, we can use nickels and pennies. The possible combinations are as follows:

- 5 nickels: No pennies needed.
- 3 nickels: Add 1 penny to the remaining 5 cents.
- 1 nickel: Add 2 pennies to the remaining 10 cents.

Case 3: Two dimes are used
In this case, the remaining amount to be made is 20 cents. The possible combinations are limited:

- 2 nickels: No pennies needed.
- 0 nickels: Add 1 penny to the remaining 5 cents.

Case 4: Three dimes are used
In this case, the remaining amount to be made is 10 cents. The only possible combination is:

- 1 nickel: No pennies needed.

Case 5: Four dimes are used
In this case, the remaining amount to be made is 0 cents. The only possible combination is:

- 0 nickels: No pennies needed.

Adding up all the combinations, we find that there are a total of 14 ways to make 40 cents using dimes, nickels, and pennies.