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Math Help - Permutation and Combination

  1. #1
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    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!
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  2. #2
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    Here is what i thought.

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

    Hope it helps.
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  3. #3
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    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
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  4. #4
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    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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