? asked in Science & MathematicsMathematics · 1 decade ago

∑k(k-1)C(n,k) using the binomial theorem and some calculus, evaluate?

∑k(k-1)C(n,k) using the binomial theorem and some calculus, evaluate

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  • kb
    Lv 7
    1 decade ago
    Favorite Answer

    By the Binomial Theorem,

    (1 + x)^n = sum(k=0 to n) C(n,k) x^k.

    Differentiating both sides twice, we get

    n(n - 1) (1 + x)^(n - 2) = sum(k=2 to n) k(k - 1) C(n,k) x^(k - 2).

    Now, set x = 1:

    n(n - 1) 2^(n - 2) = sum(k=2 to n) k(k - 1) C(n,k).

    I hope this helps!

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