Permutations and CombinationsMCQMTP June 24 Series IIIQuestion 1750 of 251
All Questions

A polygon has 44\displaystyle 44 diagonals then the number of sides are

Options

A6\displaystyle 6
B7\displaystyle 7
C8\displaystyle 8
D11\displaystyle 11
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Correct Answer

✅ Option d — 11\displaystyle 11

All Options:

  • A6\displaystyle 6
  • B7\displaystyle 7
  • C8\displaystyle 8
  • D11\displaystyle 11

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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=n(n−3)2D = \frac{n(n-3)}{2}
Given that the polygon has 44\displaystyle 44 diagonals, we can set up the equation:
n(n−3)2=44\frac{n(n-3)}{2} = 44
  ⟹  n(n−3)=88\implies n(n-3) = 88
This is a quadratic equation:
n2−3n−88=0n^2 - 3n - 88 = 0
  ⟹  n2−11n+8n−88=0\implies n^2 - 11n + 8n - 88 = 0
  ⟹  n(n−11)+8(n−11)=0\implies n(n-11) + 8(n-11) = 0
  ⟹  (n−11)(n+8)=0\implies (n-11)(n+8) = 0
Since the number of sides n\displaystyle n must be a positive integer:
n=11n = 11
Let us verify for n=11\displaystyle n = 11:
D=11×(11−3)2=11×82=44D = \frac{11 \times (11-3)}{2} = \frac{11 \times 8}{2} = 44
This matches the problem description exactly.

Hence, **Option D** 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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