Number Of Diagonals
The total number of diagonals that can be drawn in a convex polygon having 12 sides, counted using combinations of vertices, equals which value?
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About This Question
- Subject
- mathematics
- Chapter
- permutations and combinations
- Topic
- number of diagonals
- Difficulty
- Easy
- Year
- 2025
Solution
Correct Answer:
54
A diagonal joins two non-adjacent vertices of a polygon, so the count is the number of vertex pairs minus the sides, a tidy JEE Advanced combinatorial identity. A 12-sided polygon has 12 vertices, and the number of line segments joining any two vertices is C(12, 2) = 66. Of these, 12 segments are the sides of the polygon, which are not diagonals. Subtracting gives 66 - 12 = 54 diagonals. Equivalently, the diagonal formula n(n - 3)/2 = 12·9/2 = 54 confirms the result. Option 66 counts all vertex-joining segments, including the sides. Option 12 counts only the sides. Option 60 subtracts an incorrect number of sides. Hence there are 54 diagonals. Plausibility check: each vertex connects by diagonals to all but itself and its two neighbours, giving n - 3 = 9 diagonals per vertex, and 12·9/2 = 54 after halving for double counting, matching the combination approach.
This easy difficulty mathematics question is from the chapter permutations and combinations, covering the topic of number of diagonals. It appeared in the 2025 exam.
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