1. ## Permutation and Combination

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!

2. Here is what i thought.

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

Hope it helps.

3. 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

4. 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, $\displaystyle X$ and $\displaystyle Y$ must sit on the same side?

Suppose $\displaystyle X$ and $\displaystyle Y$ 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 $\displaystyle 5!$ ways.
There are: .$\displaystyle 4\cdot3\cdot5! \:=\:1440$ ways.

Suppose $\displaystyle X$ and $\displaystyle Y$ 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 $\displaystyle 5!$ ways.
There are: .$\displaystyle 3\cdot2\cdot5! :=\:720$ ways.

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

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# a table has 7 seats 4 being on one side

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