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Re-arranging logarithms

  1. Aug 23, 2011 #1
    1. The problem statement, all variables and given/known data

    Show that ln[ (N + M - 1)! /M! (N-1)! ] is equal to N ln((N+M) / N) + M ln((N+M) /M).

    2. Relevant equations

    Using stirling's formula ln N! ~ N lnN - N

    3. The attempt at a solution

    ln[ (N + M - 1)! /M! (N-1)! ] (a)
    = (N+M -1) ln(N+M -1)- (N+M -1) - M lnM + M - (N-1) ln(N-1) + (N-1) (b)
    =(N+M) ln(N+M) - M lnM - N lnN (c)
    = N ln( (N+M)/N) + M ln ( (N+M) / M) (d)

    I understand how to get from (a) to (b), and (c) to (d). But I don't understand what happens from (b) to (c). What has happened to the -1 values?

  2. jcsd
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