How many different 8 letter words are possible using the letters of the word SYLLABUS?

Analysis

To solve this problem, we need to determine the number of distinct arrangements (permutations) of the letters in the word SYLLABUS.

Concept: Permutation of Multisets

When arranging objects where some are identical, we use the formula for permutations of a multiset:

Where:

• $n$ is the total number of items.
• $n_1, n_2, \dots$ are the frequencies of the repeating items.

Step-by-Step Solution

1. Count the total number of letters ($n$): The word S-Y-L-L-A-B-U-S has 8 letters. So, $n = 8$.

2. Identify the repeating letters: • S appears 2 times. • Y appears 1 time. • L appears 2 times. • A appears 1 time. • B appears 1 time. • U appears 1 time.

3. Apply the formula: Since 'S' repeats twice and 'L' repeats twice, we divide the total factorial ($8!$) by the factorial of the frequency of each repeating letter: $$\text{Number of arrangements} = \frac{8!}{2! \times 2!}$$

Conclusion

The calculation $\frac{8!}{2! 2!}$ accounts for the fact that swapping the two 'S's or the two 'L's does not create a new unique word. This matches option C.

Exam Tip

Whenever you see a word arrangement question, always write out the letters and their counts first. If no letters repeat, the answer is simply $n!$. If letters repeat, always remember to divide by the factorial of the number of times each letter repeats!