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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

But try the problem first…

Source

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

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