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 $n$, consider any two members of $S$ that are friends. How many other members of $S$ will be common friends of both these members?
- $\frac{1}{2} (n^2 – 5n +8)$
- $2n-6$
- $\frac{1}{2} n(n – 3)$
- $n-2$
- $\frac{1}{2} (n^2 – 7n + 16)$