Say we were to uniformly sample $k$ times from a bit string with length $n$. What is the expected number of pairs with a Hamming distance $d$? In the limit of Hamming distance 0, I realize this becomes a birthday collision problem, but I am interested in the general case.
For example, say we generate 1000 random 10-bit strings, what is the expected number of strings with a Hamming distance of 3?
I found several related posts: Probability that a set of 'N' random binary strings are all at least a certain Hamming distance 'k' apart , birthday problem near misses , and Average number of strings with edit distance exactly 2 but I could not quite make the additional leap to answer my similar but not exactly the same question.
Thanks in advance for your help!