Urns (noch nicht ├╝bersetzt)

Problem 783

Given $n$ and $k$ two positive integers we begin with an urn that contains $kn$ white balls. We then proceed through $n$ turns where on each turn $k$ black balls are added to the urn and then $2k$ random balls are removed from the urn.

We let $B_t(n,k)$ be the number of black balls that are removed on turn $t$.

Further define $E(n,k)$ as the expectation of $\displaystyle \sum_{t=1}^n B_t(n,k)^2$.

You are given $E(2,2) = 9.6$

Find $E(10^6,10)$. Round your answer to the nearest whole number.