This drawer puzzle depends on what we mean by a matching pair. The usual interpretation is that the 10 socks form 5 distinct pairs.
In the worst case, you could first pull out one sock from each of the five different pairs. After taking 5 socks, you might still have no complete pair at all.
The very next sock changes that. Once you take a 6th sock, it must match one of the five socks you already selected, because there are only five pairs in the drawer.
So the minimum number needed to guarantee at least one matching pair is 6.
Answer: 6 socks.
The trick is to think about the worst possible sequence of draws, not the most likely one. You are guaranteeing a pair, so you must allow for maximum bad luck first.
