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To solve this problem, we need to calculate the total number of possible serial numbers that can be created with the given constraints:

1. **Letters Selection (2 letters):**
   - We have 6 letters (A, B, C, D, E, F) and need to choose 2 letters.
   - There can be no repeats, so once a letter is chosen, it cannot be chosen again.
   - For the first letter, we have 6 options, and for the second letter, we have 5 remaining options.
   - Therefore, the number of ways to choose the letters is:
     \[
     6 \times 5 = 30
     \]

2. **Digits Selection (4 digits):**
   - We have 10 digits (0 through 9) and need to choose 4 digits.
   - There can be no repeats among the digits.
   - For the first digit, we have 10 options. For the second digit, 9 options remain, then 8 options for the third digit, and 7 options for the fourth digit.
   - The number of ways to choose the digits is:
     \[
     10 \times 9 \times 8 \times 7 = 5040
     \]

3. **Total Serial Numbers:**
   - We multiply the number of ways to choose the letters by the number of ways to choose the digits:
     \[
     30 \times 5040 = 151200
     \]

Therefore, the total number of unique serial numbers that can be generated is **151,200**.

Would you like further details or have any questions?

Here are five related questions that build on this concept:

1. How would the answer change if the letters could be repeated?
2. What if the digits were chosen from a different set, such as only odd numbers?
3. How would the calculation change if both letters and digits could repeat?
4. What if the format changed to three letters followed by three digits?
5. How many unique serial numbers can be generated if the digits are in descending order?

**Tip:** Remember that when there are no repeats allowed, the number of choices decreases with each selection.

Serial numbers for a product are to be made using 2 letters followed by 4 digits. The letters are to be taken from the first 6 letters of the alphabet, with no repeats. The digits are taken from the 10 digits (0, 1, 2, ..., 9), with no repeats. How many serial numbers can be generated?

Learn how to calculate the number of unique serial numbers using permutations and counting principles. This problem demonstrates how to arrange 2 letters from 6 without repeats and 4 digits from 10 without repeats. Suitable for high school students in Grades 9-12.

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