a How many ways are there of seating all the guests at the t

(a) How many ways are there of seating all the guests at the table?

(b) How many ways can the guests be seated so that none of the couples are seperated (non-adjacent)?

Hint: A circular table is different from a bench in the following sense: any cyclic rotation of a configuration gives rise to the same configuration.

Solution

A. In Circular table, we have to fix the position of the first person, then we can arrange rest of persons by using the same method as used in arranging in line.

Now there are n couple, means total 2n people.

fix a person at any chair, now remaining peoples are = 2n - 1

Now we have to arrange (2n - 1) in rest of chairs, so

No. of ways = (2n - 1)!

B.

Now total no. of couple = n

Fix the position of one couple, after that remaining couples are = (n - 1)

Now we have to arrange (n - 1) couples in a line, so

Total no. of ways = (n - 1)!

Now each couple can be arranged in 2! ways, So

Total no. of ways = (n - 1)! * (2!)^n = (n - 1)! * 2^n