permutations

[carebear]

There are 11 chairs in a row. In how many ways can 5 people be seated if they sit in consecutive chairs?

I believe the answer is 5! multiplied by 7 positions to get a total of 840 ways.....first of all, is that correct?

Second, to arrive at that answer I "sketched out" the 7 possible scenarios. Is there a way to calculate it without drawing all of that out. I am worried that my teacher may ask something like......There are 100 chairs in a row. In how many ways can 4 people be seated in they sit in consecutive chairs?

The empty positions is "throwing me off"......please help.....thanks!

 

[soroban]

Hello, carebear!

There are 11 chairs in a row.
In how many ways can 5 people be seated if they sit in consecutive chairs?

I believe the answer is 5! multiplied by 7 positions to get a total of 840 ways . . . first of all, is that correct? . Yes!

Second, to arrive at that answer I "sketched out" the 7 possible scenarios.
Is there a way to calculate it without drawing all of that out.
I am worried that my teacher may ask something like:
. . There are 100 chairs in a row. In how many ways can 4 people be seated in they sit in consecutive chairs?

You can reason it out with a simpler "drawing".

Number the chairs, in order, from 1 to 100.

The four people can occupy seats: .\(\displaystyle (1,2,3,4),\;(2,3,4,5),\;(3,4,5,6),\:\hdots\;(97,98,99,100)\)

You can see that there are 97 groups of chairs that can be filled.

For each group of chairs, the four people can be permuted in 4! ways.


Therefore, there are: .\(\displaystyle 97 \times 4! \:=\:2328\) possible seatings.