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## Problem – Gaps in Permutation (ISI Entrance)

Find in how many ways can letters of the word PESSIMISTIC be arranged such that no two S and no two I can come together along with S and I cannot come together.

- 0
- 20
- 120
- 2400

**Key Concepts**

Permutations

Combinatorics

Number Theory

## Check the Answer

But try the problem first…

Answer: 2400

Source

Suggested Reading

ISI Entrance, TOMATO Objective, Problem 145

Combinatorics by Brualdi.

## Try with Hints

First hint

Arranging PESSIMISTIC generally in

$\frac{11!}{3!3!}$ ways

Second Hint

The letters P E M T C can be arranged in 5!=120 ways

Final Step

Remaining 6 slots with six letters 3 S and 3 I can be arranged in $\frac{6!}{3!3!}$=20 ways. Then number of ways =2400

Watch the video

## Other useful links

- https://www.youtube.com/watch?v=4qbh7mC6YCY&t=135s
- https://www.cheenta.com/geometry-of-tangents-isi-entrance-b-stat-2009/