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Math Help - Combinatorics - n straight lines?

  1. #1
    Super Member fardeen_gen's Avatar
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    Combinatorics - n straight lines?

    Out of n straight lines whose lengths are 1, 2, 3,...,n inches respectively. Prove that the number of ways in which four may be chosen which will form a quadrilateral in which a circle may be inscribed is \frac{1}{48}\{2n(n - 2)(2n - 5) - 3 + 3(-1)^n\}.
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  2. #2
    Senior Member
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    A restatement

    A quadrilateral with side lengths a,b,c,d can be inscribed by a circle if and only if a+c=b+d. So to restate the problem in much simpler form, given the set of numbers 1,2,3,4,5,...n, in how many ways can four distinct numbers be chosen from this set satisfying a+b=c+d without regard to order?

    Denote the sequence in question by a_n which we are conjecturing is 1,3,7,13,22,34,50,...
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