∑{k=1,n} [(-1)^k·( n C k) ]?

Question

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}

Answer 1

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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.

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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!