Permutations and CombinationsMCQPYQ Sep 24Question 1705 of 251
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In a party every person shakes hands with every other person. If there are 105 handshakes in total, find the number of persons in the party.

Options

A15
B14
C21
D22
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Correct Answer

✅ Option a — 15

All Options:

  • A15
  • B14
  • C21
  • D22

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Detailed Solution & Explanation

Let n\displaystyle n be the number of persons in the party.
A handshake takes place between exactly 2 persons.
The total number of handshakes among n\displaystyle n persons is given by the combination formula nC2\displaystyle ^nC_2:
nC2=105^nC_2 = 105
n(n−1)2=105\frac{n(n-1)}{2} = 105
n(n−1)=210n(n-1) = 210
We need to find two consecutive positive integers whose product is 210. Since 15×14=210\displaystyle 15 \times 14 = 210, we have:
n=15n = 15
Thus, there are 15 persons in the party.
Hence, **Option A** 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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