Meeting in a Club

Question said that no 'pair' have common friends pair indicates that there are total people in even number so option 4 is correct

NOTE - Before reading the solution , you should know that in the solution we have taken case of a single person that would be same for all members )Let the friends of each person be y and the total no of members be x so the no of Non friends of the person is x - y - 1 . We know that the non friends of the person have 2 friends common with the person so all the non friends are connected with the person by the y friends the person has , the non friends of the person may be friends of the friends of the person but not necessarily all. Now for each of y friends let the no. of non friends of the person that are friends of the each of the y friends of the person be z or in simpler language the total of friends of anyone of the y friends that are non friends of the person is suppose z. Now as clearly there should equal friends of each of members so the total no of friends that each of the y friends of the person have ( z + 1 , as the person is also the friend of each y friends) should be equal to the friends of the person which we have taken to be y so, z = y - 1 . Now we know that each of the non friends the person have is the friend of any two of the y friends the person has so by a little usage of mind we conclude the total non friends of the person = ( y - 1 ) y / 2 as the product of ( y -1), the no. of friends of each of y friends of the person that are non friends of the person and y , the no.of friends of the person gives the twice the the no. of non friends that the person has ( REMEMBER - each non friends has two common friends with the person ). So the total no of members = [( y - 1 ) y / 2 ] +y + 1. Now by substituting various values of y we get 16 after substituting 5 to get the correct pair 16,5 . A bit complex solution, hope it would though be helpful.