Given a group of n1 ladies and n2 gentlemen and a large roun

Given a group of n1 ladies and n2 gentlemen and a large round table with exactly n1 + n2 pitches/places.
For an unknown reason I want to make a seating arrangement such that it does not prevent a person (sir/madame) sits between two ladies.

question:

(C) Give an expression for the number of possible table arrangements for the values
of n for which this is possible. The host and hostess have a permanent place,
side by side, and that all men and women are recognizable.

Solution

Hi, I am Waqar. Below is the solution of above problem:

We will solve this by Permutation & Combination,

Since n₁ is the number of ladies & n₂ is the number of men

=> n₂ Men can sit in a round table in (n₂-1)! ways.
=>Now there are n₂ spaces in between the n₂ Men.

=> If no 2 Ladies sit together then the n₁ Ladies have to sit in these n₂ blank spaces.

=> Which can be done in (n₁₎! ways.
=> Therefore, total number of methods of arranging Men and n₁ in a round table such that no 2 Ladies sit together is (n₂-1)! x (n₁)!

Happy to help you

Waqar

Happy Chegging