The word MISSISIPPI has 10 letters, with repeated letters I, S, and P. In how many distinct permutations can all the letters of the word MISSISIPPI be arranged?

Introduction / Context:
This problem is about counting permutations of letters when some letters are repeated. When all letters are distinct, we simply use n! arrangements. However, repeated letters cause overcounting because swapping identical letters does not create a new distinct arrangement. Such problems appear frequently with words like MISSISSIPPI, BALLOON, and so on in combinatorics and aptitude tests.

Given Data / Assumptions:

• Word: MISSISIPPI.
• Total letters: 10.
• Letter counts: M appears 1 time, I appears 4 times, S appears 3 times, P appears 2 times.
• All permutations use all 10 letters at once.
• Two arrangements that look identical when written are considered the same arrangement.

Concept / Approach:
The formula for distinct permutations of a multiset with repeated elements is:
Total permutations = n! / (n1! * n2! * n3! * ...) where:

• n is the total number of letters.
• n1, n2, n3, ... are the frequencies of each distinct letter.

For MISSISIPPI, n = 10, with counts 4, 3, 2, and 1 for I, S, P, and M respectively.

Step-by-Step Solution:
Step 1: Compute the total factorial of letters: 10!. Step 2: Identify repeated letter counts: I = 4, S = 3, P = 2, M = 1. Step 3: Write the formula: total distinct permutations = 10! / (4! * 3! * 2! * 1!). Step 4: Compute 10! = 3628800. Step 5: Compute 4! = 24, 3! = 6, 2! = 2, and 1! = 1. Step 6: Multiply the denominator: 24 * 6 * 2 * 1 = 288. Step 7: Divide 3628800 by 288 to get 12600.

Verification / Alternative check:
We can simplify division stepwise: 3628800 / 24 = 151200, then 151200 / 6 = 25200, then 25200 / 2 = 12600. This confirms that all cancellations are performed correctly. Also, if all letters were distinct, we would have 10! = 3628800 arrangements, so having 12600 distinct permutations after accounting for repetitions is reasonable, because many permutations collapse to the same visual arrangement when identical letters are swapped.

Why Other Options Are Wrong:
12400 and 11160: Both are close but smaller than 12600 and arise from incorrect partial division or forgetting a factorial factor. 16200: Larger than the correct answer and suggests incomplete adjustment for repetition. Only 12600 matches the correct multiset permutation formula and accurate computation.

Common Pitfalls:
A common error is to ignore repeated letters entirely and just compute 10!, which greatly overestimates the count. Another mistake is to divide by only some of the repeated factorials, for example by 4! and 3! but forgetting 2!. Finally, mixing up the exponents or miscounting the number of occurrences of each letter leads to wrong denominators. Carefully counting each letter and applying the full formula avoids these problems.

Final Answer:
The total number of distinct permutations of the letters in MISSISIPPI is 12600.