[louis_24]

In how many ways can 6 individuals be seated in a round table with 6 chairs?

a) suppose 2 persons wanted to be seated side by side, in how many ways can they do it?

b) in how many ways can these 6 individuals arrange themselves if 2 among them refuse to sit together?

Follow up: is the problem a permutation or a combination? Thanks for the answer!

[galactus]

in how many ways can 6 individuals be seated in a round table with 6 chairs?

The are (n-1)! Ways to arrange n people around a circular table.

a) suppose 2 persons wanted to be seated side by side, in how many ways can they do it?

Tie the two people together and pretend they are one person. There are now 5 'people' instead of 6. These can all be arranged in 4! Ways.
But, don't forget that the two people next to each other can be arranged in 2 ways. Thus, there are ways.

b) in how many ways can these 6 individuals arrange themselves if 2 among them refuse to sit together?

Try reasoning through this one. Seat one person at a seat. You know the other person can not sit next to them. Thus, that only leaves three chairs for this person. The others can be seated in the other seats.