A generalization of the airplane seating puzzle

When the last person looks at the pile, which socks could possibly be there? Their own socks, and the socks of the first person. That's it. Any sock belonging to any other person will have been removed from the pile when that person picked up their socks.

Therefore, the question is simply this: from a set of $k$ black and $k$ white socks, $k$ socks are picked uniformly at random (we know that $k$ socks eventually remain, and everyone is indifferent to picking black or white socks). What is the probability that $j$ white socks remain?

This is clearly answered by your formula.