number of ways to arrange five people in seven seats

[zebrafinch]

I have this question in my homework that I cannot figure out.

Jack, Bill, John, Harry, and Brent went to the movie theater. They found a row with 7 empty seats. How many different ways can the men be seated in the row?

There are 6 possabilities for each seat I think, but I do not know how to do this kind of problem. If anyone here knows how could you please help me?

 

[pka]

\(\displaystyle \frac {7!}{2!}\)
Why?

 

[soroban]

Hello, zebrafinch!

Have you been taught nothing about "counting problems"?

Jack, Bill, John, Harry, and Brent went to the movie theater.
They found a row with 7 empty seats.
How many different ways can the men be seated in the row?

We can baby-talk our way through it . . .

Jack has a choice of any of the 7 available seats; he chooses one.
Bill has a choice of any of the 6 remaining seats; he chooses one.
John has a choice of any of the 5 remaining seats; he chooses one.
Harry has a choice of any of the 4 remaining seats; he chooses one.
Breant has choice of any of the 3 remaining seats; he chooses one.

\(\displaystyle \text{Therefore, there are: }\:7 \times 6 \times 5 \times 4 \times 3 \;=\;\boxed{2,520}\text{ possible seatings.}\)

 

[zebrafinch]

What I was confused on is that if jack chooses a chair out of 7 and so on... but if they get up and choose over again then what if harry picks first this time? I do not know if that is intellagent enough to make sense??

 

[Loren]

Any one of the seven could be in the first seat. There are 7 ways for them to be seated in the first seat. That leaves 6 to be seated.
Any one of the six could be in the second seat seat. There are 7 X 6 ways for the first two seats to be occupied. It may help if you write down the different possibilities to show that given 7 people to be seated in two seats there are 42 different ways to accomplish this.
Now, there are 5 left to be seated in the next seat. So, in the first 3 seats there are 7 X 6 x 5 different ways to seat 3 out of the seven people.
etc.