Suppose we have 2017 coins, such that the probability of tossing heads with each coin is $\frac{1}{2018}$, $\frac{2}{2018}$,......... $\frac{2017}{2018}$. We toss each coin once. Find the probability that the number of heads tossed is even.
So the total number of possibilities is $2^{2017}$.
Suppose we toss 2 heads - the number of possibilities will be $$ \begin{pmatrix} 2017 \\ 2 \\ \end{pmatrix} $$ However, we would need to multiply by the probabilities as each coin is biased, so for this case we would need to multiply by
$\frac{(1)(2)+(1)(3)+...........(2016)(2017)}{2018}$
This already is a big computation to make. Doing the other possibilities with 4, 8 and 12 heads seems impossible to do by hand. Is there some key observation that I am missing here?