Question

Find the number of ways of arranging N people in a straight line, if two particular people must always be separated.​

Answer 1

Answer:

It’s the number of ways N people can be arranged in a straight line without restriction minus the number of arrangements in which the 2 people are together.

The number of ways N people can be arranged in a straight line without restriction = N!

The number of arrangements in which the 2 people are together = the number of possible positions for the left-hand person multiplied by the number of people who can be that person, multiplied by the number of possible arrangements for the other N - 2 people

= (N - 1) * 2 * (N - 2)!

= 2 (N - 1)!

So the number of ways N people can be arranged in a straight line if 2 particular people must be separated = N! - 2(N - 1)!