Alice owes Greg 500 dollars. She has in her wallet twenty 10-dollar notes, two 50-dollar
notes, and ten 100-dollar notes.
How many different ways are there for Alice to repay Greg?

Alice owes Greg 500 dollars, and she has a set of bills consisting of:

- 20 ten-dollar notes
- 2 fifty-dollar notes
- 10 one-hundred-dollar notes

We need to determine how many different ways Alice can use these notes to repay Greg exactly 500 dollars.

### Approach:
Let \( x_1 \), \( x_2 \), and \( x_3 \) represent the number of 10-dollar, 50-dollar, and 100-dollar notes, respectively, that Alice uses. The total sum equation is:

\[
10x_1 + 50x_2 + 100x_3 = 500
\]

Where:
- \( x_1 \) is between 0 and 20 (inclusive),
- \( x_2 \) is between 0 and 2 (inclusive),
- \( x_3 \) is between 0 and 5 (inclusive).

### Steps:
1. First, express \( x_1 \) in terms of \( x_2 \) and \( x_3 \):
   \[
   x_1 = \frac{500 - 50x_2 - 100x_3}{10}
   \]
   For \( x_1 \) to be valid, \( 500 - 50x_2 - 100x_3 \) must be divisible by 10, and \( x_1 \) must be within its allowed range \( 0 \leq x_1 \leq 20 \).

2. We will iterate through all possible values of \( x_2 \) and \( x_3 \), checking the corresponding value of \( x_1 \).

### Solution:
Let’s calculate how many ways Alice can repay Greg by systematically checking each possibility.There are 8 different ways Alice can repay Greg exactly 500 dollars using her available notes.

Would you like further details or have any questions?

Here are 5 related questions to expand on this topic:
1. What would change if Alice had fewer 100-dollar notes?
2. How would the number of ways change if Alice owed 600 dollars instead of 500?
3. What if Alice only had 50-dollar and 100-dollar notes—how many ways could she repay 500 dollars?
4. How would the solution change if Alice had an unlimited number of each note?
5. Could this problem be solved using combinatorics, and how?

**Tip:** In problems like these, breaking down the total sum into combinations of smaller denominations helps simplify the solution.

Alice owes Greg 500 dollars. She has in her wallet twenty 10-dollar notes, two 50-dollar notes, and ten 100-dollar notes. How many different ways are there for Alice to repay Greg?

Discover how Alice can repay 500 dollars using different combinations of 10, 50, and 100-dollar notes. This problem involves algebraic equations and combinatorial analysis. Suitable for high school students in Grades 9-12.

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