https://file.aiquickdraw.com/mathbot/file/1726928665510rchs9twe.jpg

The task in the image involves determining how many different codes can be generated using a combination of letters and numbers. The format for the code is either:

1. **Letter, Number, Letter** (e.g., A2A)
2. **Number, Letter, Number** (e.g., 1B2)

### Step-by-Step Breakdown:
- **Letters**: There are 6 letters available: A, B, C, D, E, F.
- **Numbers**: There are 4 numbers available: 1, 2, 3, 4.

### Calculating for each format:

1. **For the format "Letter, Number, Letter"**:
   - First letter: 6 possible choices.
   - Number: 4 possible choices.
   - Second letter: 6 possible choices.

   Total combinations = \( 6 \times 4 \times 6 = 144 \) combinations.

2. **For the format "Number, Letter, Number"**:
   - First number: 4 possible choices.
   - Letter: 6 possible choices.
   - Second number: 4 possible choices.

   Total combinations = \( 4 \times 6 \times 4 = 96 \) combinations.

### Total Number of Different Codes:
Adding both formats, we get the total:
\[
144 \, (\text{Letter, Number, Letter}) + 96 \, (\text{Number, Letter, Number}) = 240 \, \text{different codes}.
\]

Thus, there are **240 different codes** altogether.

Let me know if you need further details or clarifications! 

Here are 5 questions to expand on this:
1. How would the total number of codes change if there were more numbers or letters available?
2. Can the order of pressing buttons be further optimized for fewer mistakes?
3. How would adding constraints (like no repetition of letters or numbers) change the solution?
4. Could there be an advantage to using a longer code for security purposes?
5. How would you calculate combinations if the format allowed repeating the same element (e.g., "AAA" or "111")?

**Tip**: Using factorials is an efficient way to calculate the number of different arrangements when no elements are repeated!

How many different codes can be generated using a combination of letters and numbers in the formats 'Letter, Number, Letter' and 'Number, Letter, Number'?

Learn how to calculate the total number of codes using the combinations of letters and numbers in 'Letter, Number, Letter' or 'Number, Letter, Number' formats. Explore combinatorics concepts with detailed steps and examples. Suitable for middle school students.

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