In how many ways can n+1 men and n women be seated in a row under each of the following conditions:
1) no restriction by sex
There are n+1 + n or 2n+1 people. Answer (2n+1)!
2) no men in adjacent seats
The n+1 men go in the n+1 odd-numbered seats 1,3,5,7,...,n-1,n+1.
That's (n+1)! ways to seat the men. For each of those (n+1)! ways,
the n women go in the n even-numbered seats 2,4,6,8,...n-2,n.
That's n! ways to seat the women.
That's (n+1)!n!
That can also be written (n+1)n!n! = (n+1)(n!)²
3) at least two men in adjacent seats
[The answer to 1)] MINUS [the answer to 2)]
(2n+1)! - (n+1)!n!
Edwin