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CAT-2007-Previous Years Paper

Question
13 out of 25
 

Let S be the set of all pairs (Ij) 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?



A

B n – 2

C

D

E 2n – 6

Ans. B

Consider friends (a, b) and (a, c).

Their common friend can be either (b, c) or a member of the form (a, d) or (d, a) where d is different from a, b, c. d can be chosen in (n – 3) ways.

So, number of common friends = (n – 3) + 1 = n – 2.

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