How many 4 - letter codes can be formed using the first 10 letters of the English alphabet, if no letter can be repeated?

Explanation

To find how many 4-letter codes can be formed using the first 10 letters of the English alphabet (A, B, C, D, E, F, G, H, I, J) without repeating any letters, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the total number of letters available**:
We have the first 10 letters of the English alphabet: A, B, C, D, E, F, G, H, I, J.

2. **Determine the number of choices for each letter in the code**:
- For the **first letter**, we can choose any of the 10 letters. So, there are **10 choices**.
- For the **second letter**, we cannot use the letter chosen first, so we have **9 choices** left.
- For the **third letter**, we cannot use the first two letters, leaving us with **8 choices**.
- For the **fourth letter**, we cannot use the first three letters, leaving us with **7 choices**.

3. **Calculate the total number of combinations**:
We multiply the number of choices for each position:
\[ \text{Total codes} = 10 \times 9 \times 8 \times 7 \]

4. **Perform the multiplication**:
- First, calculate \(10 \times 9 = 90\).
- Next, calculate \(90 \times 8 = 720\).
- Finally, calculate \(720 \times 7 = 5040\).

5. **Conclusion**:
Therefore, the total number of 4-letter codes that can be formed is **5040**.