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Combinatorial Methods

  1. Sep 5, 2010 #1
    1. The problem statement, all variables and given/known data

    If k people are seated in a random manner in a circle containing n chairs (n>k), what is the probability that the people will occupy k adjacent chairs in the circle?



    3. The attempt at a solution

    1) The total probability of k people seated in a circle of n chairs is [ (n "choose" k) = p1 ]

    2) The amount of people occupying k adjacent chairs. We have (n-k) total unoccupied chairs. For a small chunk of the total number, (Kth Person, 1st Person, 2nd Person are all seated next to each other) or (2nd Person, 1st Person, Kth Person seated adjacently). => (n-k+2)=p2 total ways to seat adjacently (considering the two identical orders).

    3) Solution = p2*p1 = (n-k+2) / (n "choose" k)
     
  2. jcsd
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