Suppose we have the permutation of size $$$N$$$ and the numbers in it are divided in two groups, one of size $$$a$$$ and one of size $$$b$$$ = $$$N-a$$$ (for simplicity, let's say the first $$$a$$$ numbers form the first group)
We also have an(empty) array of size $$$N$$$ also divided in two groups one of size $$$c$$$ and one of size $$$d$$$ = $$$N-c$$$ (also for simplicity, let's say the first $$$c$$$ cases form the first group)
If we were to divide the $$$N$$$ numbers(all permutations are equiprobable) in the empty array, I am looking for the expected number of numbers of group a that land in a position of group c.
Turns out this is $$$a*c/N$$$ (from the tutorial of Edu 57 F).
I tried to compute it in many ways and just ended up stuck with lots factorials, N choose Ks and the EV sum; sum of i * (probability that we get i numbers)
So I would very much appreciate any help regarding the intuition behind this and the ways of deriving such EV formulas.
Thank you :)