The derangement problem is like, "There are n people each having a hat. All the n hats are put in a box. Suppose each person takes one hat randomly from the box. What is the probability that no one gets his/her own hat." Then, what if those n people each have two hats and would take two hats randomly from the box? what is the probability that everyone gets their own two hats from the box?
For example, "In an activity, everyone prepares k hats and puts them into a box. People take hats in turn. In the turn of a person, they take k gifts from the box uniformly at random one by one (without replacement)."
Then, if n = 6 and k = 2, what is the probability that everyone gets the two hats prepared by themself?
Can this problem be solved by: $\frac{1}{12C2 \times 10C2 \times 8C2 \times 6C2 \times 4C2 \times 2C2}$ (C is for combination)
Also, for the question "what is the probability that everyone gets at least one hat not prepared by themself?", is it just simply 1 - the former probability?
and
When k = 2 and n is unknown. In the turn of person X, if the first hat X takes is from person Y, what is the probability that the other gift X takes is also from Y?