Question

in how many ways can 4 ladies and 5 gentlemen be seated in a row so that no two ladies sit together

Answer 1

Lets make a group of 2 Men and 2 Women such that the women are between the 2 men and find their arrangements. This can be done in 4C2×3C2×2!×2!=724C2×3C2×2!×2!=72

Now the above group and rest of the 2 men and 1 woman can be arrange around a table in (4–1)!=6(4–1)!=6 ways

∴∴ Arrangements when 2 women sits together =72×6=432=72×6=432

We also need to remove cases where all 3 women sit together

Therefore Total number of arrangements =6!−4!3!−72×6=144