Which of the two solutions is correct?

The first situation: there are only four situations when touching two at will, that is, white, black, white and black, and black and white.

So the probability of a pair of black socks is 1/4, which is incorrect.

Because: one white, one black, one black and one white * * * contains four situations:

White 1 black1; White 1 black 2; White 2 black1; White 2 black 2; Add white and black * * * and there are six situations.

So it is 1/6.