the variance of a binomial random variable: var(X) = E[X^2] - (E[X])^2
= E[X(X-1)] + E[X] - (E[X])^2
= n*(n-1)*P^2 + n*p - (n*p)^2
= n*p - n*p^2
now, set n = 1000, take derivative with respect to p, set equal to zero, solve for p, and plug into var(x) equation to solve for the maximum value.
i end up getting p = .5 so max var(X) = 250 (when X is a binomial random variable with parameter 1000)
