(related to Proposition: Sum of Binomial Coefficients)
The closed formula for the sum of binomial coefficients. \[\sum_{k=0}^n \binom nk=2^n\] follows immediately from the binomial theorem. \[\sum_{k=0}^n{n\choose k}x^{n-k}y^k=(x+y)^n\] by setting \(x=y=1\).