Permutation and Combination

**Tangera** · October 21st 2008, 06:27 AM

Q: A rectangular table has 7 secured seats, 4 being on one side facing the window and 3 being on the opposite side. In how many ways can 7 people be seated at the table if 2 people, X and Y must sit on the same side?

The answer I was given is 2160, but I don't know how to derive it...>.< Thank you for helping!

**tester85** · October 21st 2008, 06:46 AM

Here is what i thought.

(4P2 x 5P5) + ( 3P2 X 5P5 ) = 2160

Hope it helps.

**jaydee323** · October 21st 2008, 06:53 AM

if X picked on four seat side then
4 choices for x* 3 choices for y ( has to be on same side)*P(5,5)=1440

if X picked on three seat side then

3 choices for x* 2 choices for y * p(5,5) = 720

Total arrangements 1440+720 = 2160

**Soroban** · October 21st 2008, 07:14 AM

Hello, Tangera!

A rectangular table has 7 secured seats, 4 on one side, 3 on the other. In how many
ways can they be seated at the table if 2 people, $X$ and $Y$ must sit on the same side?

Suppose $X$ and $Y$ sit on the 4-side of the table.
. . X has 4 choices of seats; Y has 3 choices of seats.
. . The other five can be seated in $5!$ ways.
There are: . $4\cdot3\cdot5! \:=\:1440$ ways.

Suppose $X$ and $Y$ sit on the 3-side of the table.
. . X has 3 choices of seats; Y has 2 choices of seats.
. . The other five can be seated in $5!$ ways.
There are: . $3\cdot2\cdot5! :=\:720$ ways.

Therefore, there are: . $1440 + 720 \;=\;{\color{blue}2160}$ seating arrangements.

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