Find the number of permutations of 6 students sitting around a round table.

(a) In how many of these arrangements are three of the students sit together ?

(b)  In how many of the arrangements, three of the students do not sit all together ?

Number of arrangements for 6 students to sit around a round table = (6 - 1)! = 5! = 120.
(a) Tie the three students.
     Number of arrangements = 
Mix with the remaining to give a toal of 3 + 1 = 4
Seat them around a round table.
∴   The number of permutations = (4 - 1)! = 3! = 6.
Hence, the number of arrangements in which the 3 students sit together = 6 x 6 = 36
                                                                                [By using (i) and (ii)]
(b)  The number of arrangements in which the 3 students do not sit all together = 120 - 26 = 84.