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 * 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!