Question

Five people are going to be seated in a row of seven chairs. How many different ways can they be seated?

Answer 1

`color(green)(2520)`

Explanation:

There are `_7C_5 = (7!)/(5! * 2!) = 21` ways of choosing which `5` chairs from the `7` will have people sitting in them.

For each of these ways there are:
`color(white)("XXX")5` ways to pick the first person;
`color(white)("XXX")4` ways to pick the second person;
#color(white)("XXX")etc. for #5!=120# ways of arranging the people in the chosen seats.

Therefore there are
`color(white)("XXX")21xx120=2520` ways of making this seating