(related to Proposition: Alternating Sum of Binomial Coefficients)
The closed formula for the alternating sum of binomial coefficients. \[\sum_{k=0}^n (-1)^n\binom nk=0\] follows immediately from the binomial theorem. \[\sum_{k=0}^n{n\choose k}x^{n-k}y^k=(x+y)^n\] by setting \(x=1\) and \(y=-1\).