6 cards, labeled 1,2, ...6, are randomly put in 3 different envelopes, at least one card in each.

Evaluate the probability of 2,2,2 distribution.

This is the same as randomly putting in 3 cards where an envelope is allowed to be empty, and evaluating the probability that the distribution is 1,1,1.

Put the first card in a random envelope. There is a 2/3 probability that the 2nd card goes in an empty envelope, and a 1/3 probability that the 3rd card goes in an empty envelope.

So, the answer is (2/3)*(1/3) = 2/9.