How many distinct 4-letter words can be formed from the word "books"?

To determine how many distinct 4-letter words can be formed from the word "books," we need to account for both the number of letters in the word and any repetitions of letters.

The word "books" contains the following letters:

- \( b, o, o, k, s \)

So the letter "o" repeats twice. We will consider two cases:

### Case 1: All 4 letters are distinct
Here, we choose 4 distinct letters from \( b, o, k, s \), ignoring the extra "o." We have 4 distinct letters: \( b, o, k, s \). The number of distinct 4-letter words that can be formed by rearranging these is the number of permutations of 4 distinct letters:

\[
4! = 24
\]

### Case 2: Two of the letters are the same (both "o"s)
In this case, we are forming a word with 4 letters where 2 of them are "o." We select 2 other letters from \( b, k, s \). We can do this in \( \binom{3}{2} \) ways (since there are 3 letters to choose from and we need 2), and for each selection of letters, we arrange them in \( \frac{4!}{2!} \) ways (because of the repeated "o"s).

- Choose 2 letters from \( b, k, s \): \( \binom{3}{2} = 3 \)
- For each choice, arrange the letters: \( \frac{4!}{2!} = \frac{24}{2} = 12 \)

Thus, the total number of 4-letter words with two "o"s is:
\[
3 \times 12 = 36
\]

### Total Number of Distinct 4-Letter Words
Adding the two cases together, we get the total number of distinct 4-letter words:

\[
24 \, (\text{from Case 1}) + 36 \, (\text{from Case 2}) = 60
\]

So, the total number of distinct 4-letter words that can be formed from the word "books" is **60**.

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Would you like further explanation or details on any part of this solution?

Here are 5 related questions to consider:
1. How many distinct 3-letter words can be formed from the word "books"?
2. What if the word had no repeated letters, how would that change the calculation?
3. How many 5-letter words can be formed from the letters of "books"?
4. What is the general formula for calculating permutations with repeated elements?
5. How many distinct 4-letter words can be formed if all letters must be distinct?

**Tip:** When dealing with repeated elements in permutations, always divide by the factorial of the number of repetitions to avoid overcounting.

How many distinct 4-letter words can be formed from the word 'books'?

Discover how to calculate the number of distinct 4-letter words that can be formed from the word 'books,' considering repeated letters. This combinatorics problem introduces key concepts like permutations, combinations, and handling repetition. Suitable for students in Grades 10-12.

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