Suppose we have a deck of 144 cards: 36 kinds of card(A-Z,0-9) and each kind has exactly four identical cards, so 36x4=144.
Now six cards are randomly dealt, without replacement; three are dealt face-up and another three face-down. What's the probability that there is exactly one shared category between face-up cards and face-down cards?
Example1: the configuration U: AAA, D: ABB satisfies the criterion because there is exactly one shared category(in this case it is A) between face-up cards and face-down cards
Example2: the configuration U: 012, D: AD0 satisfies the criterion because there is exactly one shared category 0
Example3: the configuration U: 002, D: A00 satisfies the criterion because there is exactly one shared category 0.
Example4: the configuration U: ARZ, D: ARK does NOT satisfy the criterion because there are two shared categories A, R instead of one
Example5: the configuration U: AAA, D: 999 does NOT satisfy the criterion because `there is zero shared category instead of one
EDIT: added that the cards are dealt without replacement and added the tag conditional-probability, because the solution to the problem will probably involve it