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Math Help - Urgent Help needed!

  1. #1
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    Urgent Help needed!

    Hi Can someone plz help me with these 2 questions. Its important.

    Q1) The number 1,2....n are arranged in random order. Find the probability that the digits (a) 1 and 2, (b) 1,2 and 3, appear as neighbors in the order named.

    Q2) If n balls are placed at random into n cells, find the probability that exactly one cell remains empty.

    Thank you
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  2. #2
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    Quote Originally Posted by mohit.choudhary View Post
    Q1) The number 1,2....n are arranged in random order. Find the probability that the digits (a) 1 and 2, (b) 1,2 and 3, appear as neighbors in the order named.

    Q2) If n balls are placed at random into n cells, find the probability that exactly one cell remains empty.
    These are two curious problems. In that the first is really trivial, whereas the second is really more difficult.

    Q1a) Think of \boxed{12} as one unit.
    Then \boxed{12},3,4, \cdots ,n are n-1 objects that can be arranged in (n-1)! ways.
    Now you do part b).

    Q2) The difficulty with this question is find an adequate model.
    I think that we can conceive this as putting n identical balls into n distinct cells.
    There are \binom{2n-1}{n} ways to put n identical balls into n distinct cells.
    If we choose one cell to be empty then we must choose one of the remaining to contain two balls.
    That can be done in n(n-1) ways.
    So we get \frac{n(n-1)}{\binom{2n-1}{n}}
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