Closed form of this binomial expression?

Question

Does a closed form for this binomial expression exists?

$\sum_{K=2}^{N}\binom{N}{K}P^{K}(1-P)^{N-K}$

Thank you.

Answer 1

If the sum ran from $K=0$ to $K=N$, you’d simply have $\big(P+(1-P)\big)^N=1$ by the binomial theorem. It’s missing the first two terms, so it must be equal to

$$1-\binom{N}0P^0(1-P)^N-\binom{N}1P(1-P)^{N-1}=1-(1-P)^N-NP(1-P)^{N-1}\;.$$