CAT 2007 — QA Question 5
Answer the next 2 questions based on the information given below.
Let S be the set of all pairs (i, j) where 1 ≤ i < j ≤ n and n ≥ 4. Any two distinct members of S are called “friends” if they have one constituent of the pairs in common and “enemies” otherwise. For example, if n = 4, then S = {(1, 2), (1, 3), (1, 4), (2, 3), (2, 4), (3, 4)}. Here, (1, 2) and (1, 3) are friends, (1, 2) and (2, 3) are also friends, but (1, 4) and (2, 3) are enemies.
For general n, consider any two members of S that are friends. How many other members of S will be common friends of both these members?
Answer & solution
- A
- B
- C
n - 2
- E
Two members are friends if they have one element in common.
Let the two members be (a, b) and (b, c)
Now, all the members having one constituent as the common element are common friends.
i.e., all members having b as one of the elements will be common friend of the above two members.
There are (n – 3) elements apart from a, b and c and hence (n - 3) such friends.
Also, one pair formed by the uncommon constituents of the two friends is a common friend i.e., (a, c)
∴ There are n – 3 + 1 = n – 2 common friends.
Hence, option (d).