Permutations and CombinationsMCQPYQ Sep 24Question 1703 of 251
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A selection is to be made for one post of Principal and two posts of Vice-Principal. Amongst the six candidates called for the interview, only two are eligible for the post of Principal, while they all six are eligible for the post of Vice-Principal. The no. of possible combinations for the selection is:

Options

A12
B4
C18
D20
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Correct Answer

✅ Option d — 20

All Options:

  • A12
  • B4
  • C18
  • D20

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Detailed Solution & Explanation

We have 6 candidates in total. We need to select:
- 1 Principal
- 2 Vice-Principals
Constraints:
- Only 2 specific candidates are eligible for the post of Principal. Let's call them {P1,P2}\displaystyle \{P_1, P_2\}.
- All 6 candidates are eligible for the post of Vice-Principal.
- Since a candidate can only hold one post, we partition the selection into two cases based on who is selected as the Principal:

**Case 1: P1\displaystyle P_1 is selected as Principal (1 way)**
- The remaining candidates available for the 2 Vice-Principal posts are the other 5 candidates (including P2\displaystyle P_2).
- Number of ways to select 2 Vice-Principals = 5C2=5×42=10\displaystyle ^5C_2 = \frac{5 \times 4}{2} = 10 ways.

**Case 2: P2\displaystyle P_2 is selected as Principal (1 way)**
- The remaining candidates available for the 2 Vice-Principal posts are the other 5 candidates (including P1\displaystyle P_1).
- Number of ways to select 2 Vice-Principals = 5C2=10\displaystyle ^5C_2 = 10 ways.

Since the two cases are mutually exclusive, we sum the ways:
Total Combinations=10+10=20\text{Total Combinations} = 10 + 10 = 20
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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