Five people are going to be seated in a row of seven chairs. How many...

Step 1
There are ${\displaystyle \_7C_5=\frac{7!}{5!\cdot 2!}=21}$ ways of choosing which 5 chairs from the 7 will have people sitting in them.
Step 2
For each of these ways there are:
5 ways to pick the first person;
4 ways to pick the second person;
etc. f or $5!=120$ ways of arranging the people in the chosen seats.
Therefore there are ${\displaystyle \text{XXX}21\times 120=2520}$ ways of making this seating