Closed form for square of binomial CDF

Question

Let $P = \sum_{k=0}^{t}\binom{N}{k}c^k(1-c)^{N-k}$

I am trying to find the closed form for the term $P-P^2$ where I need to find the square of the binomial CDF above. I looked at the closed form for the binomial CDF which can be approximated as the regularized incomplete beta function but that does not help with the calculations. Does the closed form for this even exist? If so how can it be calculated?

Answer 1

Im not sure of the result but i think it is $P-P^2=P(1-P)=\sum_{i=0}^{t}\sum_{j=0}^{N-t-1}\binom N i \binom N j c^{N+(i-j)}{(1-c)}^{N-(i-j)}$