Permutations and CombinationsMCQMTP Nov 20Question 1726 of 251
All Questions

A polygon has 14 diagonals then the number of sides are

Options

A6
B7
C8
D9
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Correct Answer

✅ Option b — 7

All Options:

  • A6
  • B7
  • C8
  • D9

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Detailed Solution & Explanation

The number of diagonals D\displaystyle D in a polygon of n\displaystyle n sides is given by the formula:
D=nC2−n=n(n−1)2−n=n(n−3)2D = ^{n}C_{2} - n = \frac{n(n-1)}{2} - n = \frac{n(n-3)}{2}
Given that the polygon has 14\displaystyle 14 diagonals, we can set up the equation:
n(n−3)2=14\frac{n(n-3)}{2} = 14
  ⟹  n(n−3)=28\implies n(n-3) = 28
This is a quadratic equation:
n2−3n−28=0n^2 - 3n - 28 = 0
  ⟹  n2−7n+4n−28=0\implies n^2 - 7n + 4n - 28 = 0
  ⟹  n(n−7)+4(n−7)=0\implies n(n-7) + 4(n-7) = 0
  ⟹  (n−7)(n+4)=0\implies (n-7)(n+4) = 0
Since the number of sides n\displaystyle n must be a positive integer:
n=7n = 7
Let us verify for n=7\displaystyle n = 7:
D=7×(7−3)2=7×42=14D = \frac{7 \times (7-3)}{2} = \frac{7 \times 4}{2} = 14
This matches the problem description exactly.

**Discrepancy Note:**
The mathematical proof shows that the number of sides is 7\displaystyle 7, which corresponds to **Option B**. The textbook answer key has a typographical error, incorrectly indicating **Option A** (6\displaystyle 6) as the correct answer. A polygon of 6\displaystyle 6 sides (hexagon) only has 9\displaystyle 9 diagonals: 6×32=9\displaystyle \frac{6 \times 3}{2} = 9.

Hence, **Option B** is the correct answer.

About This Chapter: Permutations and Combinations

Paper

Paper 3: Quantitative Aptitude

Weightage

4-6 Marks

Key Topics

Factorials, Permutations, Combinations

This chapter deals with the fundamental principles of counting. It covers factorials, circular permutations, restricted permutations, combinations, and the differences between selecting items versus arranging them.

View Official ICAI Syllabus

Exam Strategy Tip

The most common mistake is confusing 'P' (Arrangement) with 'C' (Selection). If order matters (like opening a lock), use P. If order doesn't matter (like choosing a team), use C.

Related Comparison Tables

Permutations vs Combinations: The Quantitative Aptitude Guide

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