Permutations and CombinationsPYQ May 25Question 4324 of 183
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In how many of distinct permutations of the letters in "MISSISSIPPI" when four I's do not come together?

Options

A34650
B40320
C840
D33810
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Correct Answer

✅ Option d — 33810

All Options:

  • A34650
  • B40320
  • C840
  • D33810

Detailed Solution & Explanation

First, let us calculate the total number of distinct permutations of the letters in the word \"MISSISSIPPI\".
The word contains 11 letters in total: M = 1, I = 4, S = 4, P = 2.
The total number of permutations is given by:
Total Permutations=11!1!×4!×4!×2!\text{Total Permutations} = \frac{11!}{1! \times 4! \times 4! \times 2!}
Total Permutations=399168001×24×24×2=399168001152=34650\text{Total Permutations} = \frac{39916800}{1 \times 24 \times 24 \times 2} = \frac{39916800}{1152} = 34650

Next, let us calculate the number of permutations where all four I\'s come together.
If the four I\'s always come together, we treat them as a single entity: (IIII).
The remaining letters are M (1), S (4), and P (2).
Together with the single entity (IIII), we have a total of 1+1+4+2=8\displaystyle 1 + 1 + 4 + 2 = 8 entities to arrange.
The number of permutations of these 8 entities is:
Permutations (together)=8!1!×4!×2!=4032024×2=4032048=840\text{Permutations (together)} = \frac{8!}{1! \times 4! \times 2!} = \frac{40320}{24 \times 2} = \frac{40320}{48} = 840

Finally, the number of permutations where the four I\'s do not come together is the total number of permutations minus the permutations where they are together:
Required Permutations=Total−Together\text{Required Permutations} = \text{Total} - \text{Together}
Required Permutations=34650−840=33810\text{Required Permutations} = 34650 - 840 = 33810
Hence, **Option D** is the correct answer.

About This Chapter: Permutations and Combinations

Paper

Paper 3: Quantitative Aptitude

Weightage

4-6 Marks

Key Topics

Factorials, Permutations, Combinations

This chapter deals with the fundamental principles of counting. It covers factorials, circular permutations, restricted permutations, combinations, and the differences between selecting items versus arranging them.

View Official ICAI Syllabus

Exam Strategy Tip

The most common mistake is confusing 'P' (Arrangement) with 'C' (Selection). If order matters (like opening a lock), use P. If order doesn't matter (like choosing a team), use C.

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