Flip 100 Coins

$X$ is the amount of heads. E[X] is equal to $\frac{100}{2}=50$. The expected value of the number of interest is given by \begin{equation} E[X(100-X)] = 100E[X] - E[X^2] \end{equation} Here, $E[X]$ is already known, which is 50. $E[X^2]$ has to be estimated as follows. \begin{equation} Var(X) = E[X^2] - E[X]^2=E[X^2]-50^2 \end{equation} The variance of $X$ is given by the binomial distribution. For $n=100$ and $p=0.5$, we have \begin{equation} Var(X) = n*p(1-p)=25 \end{equation} So, now we know that \begin{equation} E[X^2] = Var(X) + E[X]^2 =25+2500=2525 \end{equation} Then we can conclude that \begin{equation} E[X(100-X)] = 5000 - 2525 = 2475 \end{equation}