How many distinct permutations of the letters of the word "MESMERISE" can be formed?

Introduction / Context:
This question checks understanding of permutations of letters when some letters are repeated. Instead of simply taking factorial of the total number of letters, we must adjust for repetitions to avoid counting identical arrangements multiple times. This is a standard concept in permutations and combinations for aptitude exams and competitive tests.

Given Data / Assumptions:

• Word: MESMERISE.
• Total letters = 9.
• Letter counts: M appears 2 times, E appears 3 times, S appears 2 times, R appears 1 time, I appears 1 time.
• All permutations are considered; there is no extra restriction like vowels together or position based conditions.
• Arrangements that look the same are not counted multiple times.

Concept / Approach:
For a word with repeated letters, the number of distinct permutations is given by the formula:\nN! / (n1! * n2! * ... ), where N is the total number of letters and n1, n2, etc. are the factorials of the counts of each repeating letter. Here, N = 9. We divide by 2! for the two M's, by 3! for the three E's and by 2! for the two S's. This corrects the overcount that occurs if we treat these identical letters as distinct.

Step-by-Step Solution:
Total letters in "MESMERISE" = 9, so start with 9! permutations. Counts of repeating letters: M = 2, E = 3, S = 2. Formula for distinct permutations = 9! / (2! * 3! * 2!). Compute 2! = 2 and 3! = 6. Denominator = 2 * 6 * 2 = 24. Compute 9! = 362880. Now divide: 362880 / 24 = 15120.

Verification / Alternative check:
We can approximate 9! as 362880 and divide stepwise: first by 2 to get 181440, then by 3 to get 60480, then by 2 again to get 30240, and finally realise we must still divide by another 2 to match the denominator 24, giving 15120. This stepwise division confirms that no arithmetic mistake has been made. The final number is reasonable for a 9 letter word with multiple repetitions, because it is smaller than 9! but still quite large.

Why Other Options Are Wrong:

• 30240: This appears when you divide 9! by only 2! * 3! or miss one of the 2! factors for repeated letters.
• 7560: This is too small and would come from dividing by too large a denominator or miscounting the repetitions.
• 362880: This is 9! and ignores repetitions completely, overcounting many identical arrangements.

Common Pitfalls:
Learners often forget to account for each repeated letter or misread how many times a letter appears in the word. Another common error is to use the formula mechanically without verifying the counts of each letter, especially when a letter appears three times as E does here. Rushing the factorial arithmetic can also introduce mistakes. To avoid errors, always list letter counts clearly, write the full denominator, and then simplify slowly. Checking that the final answer is smaller than 9! but not drastically tiny helps to catch obvious issues.

Final Answer:
The number of distinct permutations of the letters of the word "MESMERISE" is 15120.