With no restrictions, 6 people can sit in 6! = 720 ways
Having two people sit together means we can treat those two as a unit, and
we get effectively 5 people to be arranged: 5! = 120 ways
BUT we must multiply this by 2! = 2 because the two-person unit can be formed in 2! ways (AB and BA): 2*120 = 240.
a) 240 ways
b) For this part, the physical configuration changes things. It is like the row A-B-C-D-E-F but curved around so F is actually understood to be next to A.
I'll try to draw it:
B - C
/ \
A D
\ /
F - E
This greatly reduces the number of permutations (seating arrangements). Here we assume it doesn't matter if, say, F is at "7 O'clock" --- if we rotated the above arrangement so F was at, say, "11 O'clock" but kept all the relative positions intact, it would still be the same arrangement.
The process:
Seat A anywhere. 'A' serves as a reference point in a way, leaving 5 seats for the others to occupy around A. Those 5 others can be arranged in 5! = 120 ways. And that is all of the ways they can be seated.
b) 120 ways
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EDIT: Oh sorry, I think you wanted the circle configuration for two people sitting together (say AB, BA)... In this case it is like before: you effectively have 5 people (with one "person" a two-person "unit") so you have 4! = 24 ways to arrange those, times 2! = 2 ways to arrange the two-person unit (AB or BA). That's 2*24 = 48 ways.