I would treat this as a multinomial with 3 groups.

I would add the probability of exactly 1, exactly 2, exactly 3 or exactly 4 times we select a 6 as the highest number.

The is the different of the two sets.

If you list all possible sets of four numbers and you first select any number from 0 to 6, but then you remove

all of those that only consist of 0 through 5 you end up with at least one 6 somewhere.