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

Oct 13, 2016

$\textcolor{g r e e n}{2520}$

#### 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:
$\textcolor{w h i t e}{\text{XXX}} 5$ ways to pick the first person;
$\textcolor{w h i t e}{\text{XXX}} 4$ 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
$\textcolor{w h i t e}{\text{XXX}} 21 \times 120 = 2520$ ways of making this seating