Let's say I am distributing a bunch of encryption keys, where if two users have the same key, they can communicate. I have $20$ different keys, and I am giving each user $2.$ Since encryption keys don't disappear when I hand them out, it can be considered 'with replacement', but I'm not giving the same user the same key twice.
I want to know what the probability that two users can communicate under these circumstances. Doing a bit of the math from what I know, each user has $20 \cdot 19 = 380$ permutations of keys. The order that they get the keys in doesn't matter, though, so it's actually $190$ different combinations of keys. So, two users have $190$ possible combinations of four keys, and what is the probability that one of their two keys is in common.
That reasoning is about as far as I get. How does one move forward from there? Thanks!