CAT 2007 | Question: 04

Question

Directions for the below question:

Let $S$ be the set of all pairs $(i, j)$ where $ 1 \leq i \leq j < n$ and $n \geq 4$. Any two distinct number 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, $(1, 2), (2, 3)$ are also friends, but $(1, 4)$ and $(2, 3)$ are enemies.

For general, $n$ how many enemies will each member of $S$ have?

• A. $n-3$
• B. $\frac{1}{2} (n^2 – 3n -2)$
• C. $2n-7$
• D. $\frac{1}{2} (n^2 – 5n +6)$
• E. $\frac{1}{2} (n^2 – 7n + 14)$

Answer 1

1) number of enemesis of (I,j)= number of pairs which don't include I or j =(n-2 , 2 )

Ans : 2

2)

number of mutual friends of (I,j) and (I,k)

is (j,k) + set of pairs which include I (other than j,k) = 1+n-3 = n-2