Let n be the number of ways that 10 dollars can be changed into dimes and quarters, with at least one of each coin being used. Then what is n?

We are tasked with determining how many ways 10 dollars can be changed into dimes and quarters, given that at least one of each coin must be used. 

### Step 1: Set up variables and equation
Let:
- \( x \) be the number of dimes.
- \( y \) be the number of quarters.

The total value of the coins must equal 10 dollars, or 1000 cents. Since each dime is worth 10 cents and each quarter is worth 25 cents, the equation that governs the situation is:

\[
10x + 25y = 1000
\]

### Step 2: Simplify the equation
We can simplify the equation by dividing through by 5:

\[
2x + 5y = 200
\]

### Step 3: Solve for \( x \)
We want to express \( x \) in terms of \( y \). Solving for \( x \):

\[
2x = 200 - 5y
\]
\[
x = \frac{200 - 5y}{2}
\]

For \( x \) to be an integer, \( 200 - 5y \) must be even. Since \( 5y \) is always odd for odd \( y \) and even for even \( y \), it follows that \( y \) must be even. Let \( y = 2k \), where \( k \) is a non-negative integer.

### Step 4: Determine the range of \( k \)
Substitute \( y = 2k \) into the equation for \( x \):

\[
x = \frac{200 - 5(2k)}{2} = \frac{200 - 10k}{2} = 100 - 5k
\]

Thus, \( x = 100 - 5k \), and \( y = 2k \).

### Step 5: Determine constraints
For both \( x \) and \( y \) to be non-negative integers, the following conditions must hold:
- \( x = 100 - 5k \geq 1 \) (since at least one dime must be used).
- \( y = 2k \geq 1 \) (since at least one quarter must be used).

From \( 100 - 5k \geq 1 \), we get \( k \leq 19.8 \), so \( k \leq 19 \).

From \( 2k \geq 1 \), we get \( k \geq 1 \).

Thus, \( k \) must be an integer between 1 and 19, inclusive.

### Step 6: Count the number of solutions
The values of \( k \) range from 1 to 19, so there are \( 19 \) possible values for \( k \).

### Final Answer:
The number of ways to change 10 dollars into dimes and quarters, with at least one of each, is \( \boxed{19} \).

Would you like further details on any of the steps?

Here are 5 related questions to extend this concept:
1. How would the solution change if nickels were also allowed?
2. What if the total amount was 20 dollars instead of 10 dollars?
3. How can this problem be solved using generating functions?
4. What is the general form of this problem if the coin values were \( a \) and \( b \)?
5. How many ways can 10 dollars be made using any number of pennies, nickels, dimes, and quarters?

**Tip:** When working with Diophantine equations, always check the conditions for integer solutions carefully.

Explore how to find the number of ways to change 10 dollars into dimes and quarters, with at least one of each coin. This problem uses Diophantine equations and algebraic techniques. Suitable for high school students.

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