∑{k=1,n} [(-1)^k·( n C k) ]?

Find the value of the summation:

∑{k=1,n} [(-1)^k·( n C k) ]

Explain.

Thanks.

Update:

And what if k begins at zero ?

∑{k=0,n}

1 Answer

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  • kb
    Lv 7
    7 years ago
    Favorite Answer

    Use the Binomial Theorem:

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

    Let x = -1:

    0 = ∑(k = 0 to n) C(n, k) (-1)^k.

    ----------------

    By writing out the 0th term of this summation, we obtain

    C(n, 0) (-1)^0 + ∑(k = 1 to n) C(n, k) (-1)^k = 0

    ==> 1 + ∑(k = 1 to n) C(n, k) (-1)^k = 0

    ==> ∑(k = 1 to n) C(n, k) (-1)^k = -1.

    I hope this helps!

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