Comprehension
Let \( S \) be the set of all pairs \((i,j)\) where \( 1 \leq i < j \leq n \) and \( n \geq 4 \). 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.
Question: 1
For general \( n \), how many enemies will each member of \( S \) have?
For general \( n \), how many enemies will each member of \( S \) have?
Show Hint
In “friend/enemy” problems, count friends first, then subtract from the total excluding self.
Updated On: Jul 31, 2025
- \( n - 3 \)
- \( \frac{1}{2}(n^2 - 3n - 2) \)
- \( 2n - 7 \)
- \( \frac{1}{2}(n^2 - 5n + 6) \)
- \( \frac{1}{2}(n^2 - 7n + 14) \)
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The Correct Option is D
Solution and Explanation
Total members in \( S \) = \(\frac{n(n+1)}{2}\).
For a given pair \((i,j)\), “friends” are those sharing \( i \) or \( j \). Count of such friends = \((n-1) + (n-1) - 1 = 2n - 3\).
Enemies = total members - 1 (itself) - friends =
\[
\frac{n(n+1)}{2} - 1 - (2n - 3) = \frac{n^2 + n - 4n + 4}{2} = \frac{n^2 - 3n + 4}{2} - 1 = \frac{n^2 - 5n + 6}{2}.
\]
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Question: 2
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?
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?
Show Hint
When counting common friends, focus on the fixed common index and exclude the other indices from both pairs.
Updated On: Jul 31, 2025
- \( \frac{1}{2}(n^2 - 5n + 8) \)
- \( 2n - 6 \)
- \( \frac{1}{2}n(n-3) \)
- \( n - 2 \)
- \( \frac{1}{2}(n^2 - 7n + 16) \)
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The Correct Option is D
Solution and Explanation
Two members are friends if they share a common index. If both share \( i \) in their pair, then the common friends are all pairs that include \( i \) but not the other member’s distinct index. This yields \( n-2 \) possibilities.
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