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Math Help - Probability Question

  1. #1
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    Probability Question

    In a game of chess there are 32 pieces (16 black and 16 white). Out of the 32 pieces there are 4 rooks, what is the chance that if all the pieces were placed in a line, no rooks are next to each other? Pieces are selected at random

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    Sssouljah
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  2. #2
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    Hello, sssouljah!

    In a game of chess there are 32 pieces, including 4 Rooks.
    If all the pieces were placed in a line in random order,
    what is the probability that no rooks are next to each other?
    There are: 2 Kings, 2 Queens, 4 Bishops. 4 Knights, 4 Rooks, and 16 Pawns.

    For simplicity, assume that all 32 pieces are distinguishable.
    . . Then there are: . 32! possible arrangements.


    Place the 28 non-Rooks in row, leaving a space before, after and between them.
    . . \_\ X\ \_\ X\ \_\ X\ \_\;\;  \hdots\;\; \_\ X\ \_\ X\ \_

    There are 28 pieces and 29 spaces.

    The 28 pieces can be arranged in 28! ways.

    Select 4 spaces for the Rooks.
    . . The Rooks can be placed in: . _{29}P_4 ways

    Hence, there are: . (28!)(_{29}P_4) ways to have the Rooks non-adjacent.


    Therefore: . \boxed{P(\text{no adjacent Rooks}) \;=\;\frac{(28!)\left(_{29}P_4\right)}{32!}}


    The answer comes out to: . \frac{819}{1240} \;\approx\;0.66

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