tammy want to get change for 1$. the only coins she can get are half dollars, quarters, and nickels. how many different ways can she get a 1$ using only these coins?
To determine how many different ways Tammy can get a 1$ using only half dollars, quarters, and nickels, we can approach the problem using a systematic method.
Since Tammy only has three types of coins available (half dollars, quarters, and nickels), let's consider each coin one by one.
1. Half dollars ($0.50): The maximum number of half dollars Tammy can use to make a 1$ is 2, as using 3 or more would exceed 1$. So, we have the following possibilities for half dollars: 0, 1, or 2 half dollars.
2. Quarters ($0.25): The maximum number of quarters Tammy can use, considering she has already used x half dollars, is (4 - 2x), as using more quarters would exceed the remaining amount for nickels. So, for each number of half dollars Tammy uses, we have the following possibilities for quarters:
- If she uses 0 half dollars, she can have 0 to 4 quarters (0, 1, 2, 3, or 4).
- If she uses 1 half dollar, she can have 0 to 2 quarters (0, 1, or 2).
- If she uses 2 half dollars, she can have 0 quarters (0).
3. Nickels ($0.05): The maximum number of nickels Tammy can use, based on the remaining amount after using x half dollars and y quarters, is (20 - 10x - 25y), as using more nickels would exceed the remaining amount. So, for each combination of half dollars and quarters, we have the following possibilities for nickels:
- If Tammy uses 0 half dollars and 0 quarters, she can have 0 to 20 nickels (0, 1, 2, 3, ..., or 20).
- If Tammy uses 0 half dollars and 1 quarter, she can have 0 to 15 nickels (0, 1, 2, 3, ..., or 15).
- If Tammy uses 0 half dollars and 2 quarters, she can have 0 to 10 nickels (0, 1, 2, ..., or 10).
- If Tammy uses 0 half dollars and 3 quarters, she can have 0 to 5 nickels (0, 1, 2, 3, or 5).
- If Tammy uses 0 half dollars and 4 quarters, she can have 0 nickels (0).
- If Tammy uses 1 half dollar and 0 quarters, she can have 0 to 15 nickels (0, 1, 2, 3, ..., or 15).
- If Tammy uses 1 half dollar and 1 quarter, she can have 0 to 10 nickels (0, 1, 2, ..., or 10).
- If Tammy uses 1 half dollar and 2 quarters, she can have 0 to 5 nickels (0, 1, 2, 3, or 5).
- If Tammy uses 2 half dollars and 0 quarters, she can have 0 to 10 nickels (0, 1, 2, ..., or 10).
- If Tammy uses 2 half dollars and 1 quarter, she can have 0 to 5 nickels (0, 1, 2, 3, or 5).
- If Tammy uses 2 half dollars and 2 quarters, she can have 0 nickels (0).
To find the total number of different ways Tammy can get a 1$, we multiply the number of possibilities for each coin. In this case, we have 3 possibilities for half dollars, various possibilities for quarters (ranging from 0 to 4 for different numbers of half dollars used), and various possibilities for nickels (depending on the combination of half dollars and quarters).
Thus, we can calculate the total number of possibilities by summing up the possibilities for each combination:
(3 half-dollar possibilities) * (5 quarter possibilities for each half-dollar possibility) * (21 nickel possibilities for each combination of half dollars and quarters).
Therefore, Tammy can get a 1$ in a total of (3 * 5 * 21) = 315 different ways using only half dollars, quarters, and nickels.