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Try this beautiful problem from the PRMO, 2019 based on Two arrangements.

## Two arrangements – PRMO 2019

Five persons wearing badges with numbers 1,2,3,4,5 are seated on 5 chairs around a circular table. Find the number of ways they be seated that no two persons whose badges have consecutive numbers are seated next to each other.

- is 107
- is 10
- is 840
- cannot be determined from the given information

**Key Concepts**

Arrangement

Algebra

Number Theory

## Check the Answer

But try the problem first…

Answer: is 10.

Source

Suggested Reading

PRMO, 2019, Question 5

Combinatorics by Brualdi

## Try with Hints

First hint

Here they may seat in circular way such that 1,3,5,2,4 and again 1

Second Hint

then they may seat in another circular way 1,4,2,5,3 and again 1

Final Step

Then number of ways = (2)(5) =10.

## Other useful links

- https://www.cheenta.com/smallest-perimeter-of-triangle-aime-2015-question-11/
- https://www.youtube.com/watch?v=ST58GTF95t4&t=140s

Google