in agroup of 5 people any two are either friends or enemies, no three of them are friends of each other and no three of them are enemies of each other. prove that every person in this group has exactly two friends.

Hi Aritra,

Let a b c d e be the 5 persons.

Say a has only 1 friend (say that is b). Now it is given that any two in the group are either friends or enemies.

So a has to be enemy with c,d,e.

Now consider c. He has to be friend with d and e (as a is already a enmy with both as well as c.... so c cannot be enemy with d,e).

Now if we consider d (enemy with a, and friend with c)... So now d cannot be enemy with e and also cannot be friend with e... clear case of contradiction. So a cannot have only 1 friend.

Now say a has three friends (b,c,d).

So b has to be enemy with c,d.... so in this case c,d cannot be either friends or enemies... contradiction

Similarly a cannot have 4 friends.

So only possibility is a has two friends and two enemies.

Similarly for all the other persons.....

Ans that solves the question.... This is more of a logical question than Permutation and Combination.

Hope it helps.

Best Regards,

Ashwin (IIT Madras).

Approved