- #1

resurgance2001

- 185

- 5

## Homework Statement

The back row of a cinema has 12 seats, all of which are empty. A group of 8 people including Mary and Francis, sit in this row.

Find the number of ways they can sit in these 12 seats if

a) There are no restrictions

b) Mary and France's do not sit in seats which are next to each other

C) All 8 people sit together with no empty seats.

## Homework Equations

nCr = n!/r! (n-r)! nPr = n! (n-r)!

## The Attempt at a Solution

a) There are no restrictions

12P8 = 19,958,400

b) This is where I get stuck. I tried to calculate the number of arrangements where Mary and Frances are sitting next to each and then subtract this from the answer to a). [/B]

**I reason that there are 11 place where the two women can sit next to each and that for each place there are two ways they can sit.**

So 11 X 2 = 22

Then the remaining six people could sit in any of the 10 remaining seats 10P6 ways

This gives a totals of 22 X 5 X 6 X 7 X 8 X 9 X 10 = 3,326, 400 which is more than I found in part a). So this has to be incorrect.

c) There are 5 X 8! ways = 321,600

Which I think might be correct.

So 11 X 2 = 22

Then the remaining six people could sit in any of the 10 remaining seats 10P6 ways

This gives a totals of 22 X 5 X 6 X 7 X 8 X 9 X 10 = 3,326, 400 which is more than I found in part a). So this has to be incorrect.

c) There are 5 X 8! ways = 321,600

Which I think might be correct.