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    Recursive relationships

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    Consider the recursive relationship for combinations:

    C(n,r) = C(n-1,r) + C(n-1,r-1)

    Prove this relationship algebraically using the mathematical definition of a combination, as well as that of the factorial function.

    Provide a logical explanation for this relationship (Hint: consider n objects as consisting of n-1 existing objects plus a new nth object. Given the ultimate goal of selecting r objects from the entire set of n, you must select objects from the set of n-1 objects and decide how the new nth object should join the existing selected sets.)

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    Solution Preview

    Because, C(n,r) = n!/(r!*(n-r)!)
    C(n-1,r) + C(n-1,r-1)
    = {(n-1)!/(r!*(n-1-r)!)} + {(n-1)!/((r-1)!*(n-r)!}
    = {(n-1)!/((r-1)!*(n-r-1)!)} * { (1/r) + (1/(n-r))}
    (Because, (n-r)! = (n-r-1)!*(n-r); r! = (r-1)!*r)

    = {(n-1)!/((r-1)!*(n-r-1)!)} * {(n-r + ...

    Solution Summary

    This solution shows a simple proof regarding recursive relationships. It is solved in step by step format.