Question

There are 2 twin sisters among a group of 15 persons. In how many ways can the group be arranged around a circle so that there is exactly one person between the two sisters?

(a) 15 *12! * 2!

(b) 15! * 2!

(c) ^14C2

(d) 16 * 15!

The question was asked in an international level competition.

I need to ask this question from Counting topic in chapter Counting of Discrete Mathematics

Answer 1

Right answer is (a) 15 *12! * 2!

For explanation I would say: We know that n objects can be arranged around a circle in \(\frac{(n−1)!}{2}\). If we consider the two sisters and the person in between the brothers as a block, then there will 12 others and this block of three people to be arranged around a circle. The number of ways of arranging 13 objects around a circle is in 12! ways. Now the sisters can be arranged on either side of the person who is in between the sisters in 2! ways. The person who sits in between the two sisters can be any of the 15 in the group and can be selected in 15 ways. Therefore, the total number of ways 15 *12! * 2!.