The number of ways in which 6 gentlemen and 3 ladies be seated round a table so that every gentlemen may have a lady by his side is

First we arrange 6 men round a table.This can be done in 5! ways.Next we arrange 3 ladies in 3 alternate gaps (out of 6 gaps between men). This can be done in 2 x 3! ways     Hence the number of ways in which 6 gentlemen and 3 ladies be seated round a table = 5! x 2 x 3! = 1440