How many four-letter words can be formed using letters from the set {A, B, C, D, E, F, G, H} if


(a) the letters can be repeated?


(b) the letters cannot be repeated?

To determine how many four-letter words can be formed from the set \(\{A, B, C, D, E, F, G, H\}\), we need to consider the scenarios where letters can be repeated and where they cannot.

### (a) Letters Can Be Repeated

If the letters can be repeated, each position in the four-letter word can be filled by any of the 8 letters. Therefore, the number of possible words is calculated by:

\[
8 \times 8 \times 8 \times 8 = 8^4 = 4096
\]

### (b) Letters Cannot Be Repeated

If the letters cannot be repeated, we need to choose a different letter for each of the four positions. The number of possible words is calculated using permutations:

- **First position:** 8 choices (any letter from the set).
- **Second position:** 7 choices (since one letter is used).
- **Third position:** 6 choices.
- **Fourth position:** 5 choices.

Thus, the total number of possible words is:

\[
8 \times 7 \times 6 \times 5 = 1680
\]

### Summary

- **(a)** Number of four-letter words with repetition: **4096**
- **(b)** Number of four-letter words without repetition: **1680**

If you have any questions or need further explanations, feel free to ask! Here are some related questions you might consider:

1. How would the solutions change if only vowels are used?
2. What if the word must start and end with a vowel?
3. How does the approach change if we want to form words of different lengths?
4. How can we extend these calculations to include a larger set of letters?
5. What if a particular letter must be included in each word?

**Tip:** Always verify whether repetition is allowed or not in permutation and combination problems to apply the correct formula.

Explore how to count the number of four-letter words that can be formed using a set of letters with and without repetition. This problem covers combinatorics, permutations, and factorials, with detailed calculations and scenarios provided.

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