Hi Avdesh.This is more of a routes and networks puzzle, so let us consider every person in the club to be a point, and their friendship is defined by the line joining the two points.Now, let us say P is a point in this network. And let P have k friends named q1, q2, ... , qk.Now, we can say that every connection of P cannot be connected to any of the points of the q series, since a pair of friends has no friends in common.Now, corresponding to every pair (qi, qj) {i, j belongs to [1 to k]}, there have to be exactly 2 points that are connected to both qi and qj. One of these is P, the other one can be represented as Rij.So, for (q1, q2), there is R12.So, the total number of points in this network = 1 + k + C(k, 2) {no. of pairs} = 1 + k + k(k - 1)/2.Now, there are at least 5 and at most 21 people in the club based on the given range.So, the possible values of k are: k = 3, k = 4, k = 5.Now if k = 3, total points = 1 + 3 + 3 = 7. So, the friends will be P, q1, q2, q3, R12, R23, and R13. But, R12 and q3 are not friends. So, there need to have 2 common friends. Making R23 and R13 friends is not possible. So, this option is out.Now, if k = 4, total points = 1 + 4 + 6 = 11. So, the friends will be P, q1, q2, q3, q4, R12, R13, R14, R23, R24, R34. But, R12 is not friends with q3 here. So, we can give only 1 common friend here, which is R34. So, this option is out as well.Thus, k = 5 and total members in the club = 16

Caselets Every Saturday, the members of Raja Harish Chandra Club meet in the evening. All the members of the club are honest and never lie. Last Saturday, the following conversation was heard at one of the tables with five members sitting around it.Satya Sadhan: In this club not all members are friends with each other.Satyabrata: None of the pair of friends in this club has any common friend.Satyajit: Every pair of members who are not friends has exactly two common friends in this club.Satya Pramod: There are fewer than 22 people in this club.Example 1: How many members are there in the club?

Caselets Every Saturday, the members of Raja Harish Chandra Club meet in the evening. All the members of the club are honest and never lie. Last Saturday, the following conversation was heard at one of the tables with five members sitting around it.Satya Sadhan: In this club not all members are friends with each other.Satyabrata: None of the pair of friends in this club has any common friend.Satyajit: Every pair of members who are not friends has exactly two common friends in this club.Satya Pramod: There are fewer than 22 people in this club.Example 1: How many members are there in the club?