Alternating Sum and Difference of Binomial Coefficients for Given n

Theorem

 * $$\sum_{i=0}^n \left({-1}\right)^i \binom n i = 0$$ for all $$n > 0$$

where $$\binom n i$$ is a binomial coefficient.

Proof 1
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We note:
 * $$\binom n {0} = \binom {n-1} {0} = 1$$ so $$\binom n {0} - \binom {n-1} {0}$$;
 * $$\left({-1}\right)^{n-1} \binom {n-1} {n-1} = - \left({-1}\right)^n \binom n n = \left({-1}\right)^n$$ so $$\left({-1}\right)^{n-1} \binom {n-1} {n-1} + \left({-1}\right)^n \binom n n = 0$$.

Hence the result.

Proof 2
From the Binomial Theorem, we have that:


 * $$\forall n \in \Z_+: \left({x+y}\right)^n = \sum_{i=0}^n \binom n i x^{n-i} y^i$$

Putting $$x = 1, y = -1$$, we get:

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