Part 2 should be 6! x 2!, not 6! x 2^6. Here's the
explanation:
Ok lets imagine 6 distinct groups first.
For the first slot, there are 6 possible combinations of groups to
put into that slot. And then, within that group, you have 2! ways
to arrange the people in the group.
For the 2nd slot, you have 5 possible groups to fit in now and
again you have 2! ways to arrange the people in that group.
Repeating this for the 4 remaining slots, the total combinations is
6 x 2! x 5 x 2! x 4 x 2! x 3 x 2! x 2 x 2! x 1 x 2! = 6! x
2!.
It's the same case for part 5 also.
