Problem Solving

2 Men, 3 Women, 5 chairs. How many ways if no two men sits together?

Total number of arrangements = 5! = 120

Now the following is the scenario in which 2 Men are together,
MM---
-MM--
--MM-
---MM

Hence picking 2M from 2M = 2C2 = 1 way and arranging will be 2 ways. So for Men it would be 2 ways totally.

Selecting 3 women would be 3C3 = 1. The 3W can be arranged in 3! ways like MMW1W2W3 or MMW2W1W3 etc totally it will 6 ways similarly for all the 4 cases it would be 6*4 = 24 ways

Total arrangement where 2M are together = 24 * 2 = 48 ways.

Required arrangement = 120 - 48 = 72 ways.

Not sure if this is correct..

-Deepak

Forget about the chairs.

All we need is to arrange 2 boys and 3 girls in a row such that boys are not together.

First we can arrange girls in 3! =6 ways

_ G2 _ G1 _ G3 _


Now the 2 boys can be seated in any of the 4 places available in 4P2 = 12 ways

Hence total number of ways = 6 x 12 = 72 ways