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Science Forum

Science Forum

  1. 04 Sep '17 18:20 / 2 edits
    why does

    ∑[g=h, ∞] ( C(h – 1, m – 1) / C(g, m) ) = m/(m – 1)
    where
    C(n, k) = n! / ( k! (n – k)! ) and is a binomial coefficient.
    g, h, m ∈ ℕ
    g≥h≥1, h≥m≥1
    ?

    I checked this with a computer program and it appears to be always correct but cannot see algebraically why it should be.
    Using C(n, k) = n! / ( k! (n – k)! )
    that can be reexpressed as

    ∑[g=h, ∞] ( (h – 1)! /( (m – 1)! (h – m)! ) / ( g! / ( m! (g – m)! ) )
    = ∑[g=h, ∞] m! (g – m)! (h – 1)! / ( g! (m – 1)! (h – m)! )
    but
    m! / (m – 1)! = m
    thus
    = ∑[g=h, ∞] m (g – m)! (h – 1)! / ( g! (h – m)! )
    = m * ∑[g=h, ∞] (g – m)! (h – 1)! / ( g! (h – m)! )
    but then why does

    ∑[g=h, ∞] (g – m)! (h – 1)! / ( g! (h – m)! ) = 1/(m – 1)
    where
    g, h, m ∈ ℕ
    g≥h≥1, h≥m≥1

    ?