Permutations and CombinationsMCQMTP June 24 Series IIQuestion 1743 of 251
All Questions

In an election, there are five candidates contesting for three vacancies; an elector can vote any number of candidates not exceeding the number of vacancies. In how many ways can one cast his votes?

Options

A12\displaystyle 12
B14\displaystyle 14
C25\displaystyle 25
DNone of these
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Correct Answer

✅ Option c — 25\displaystyle 25

All Options:

  • A12\displaystyle 12
  • B14\displaystyle 14
  • C25\displaystyle 25
  • DNone of these

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Detailed Solution & Explanation

We are given:
- Number of candidates = 5\displaystyle 5
- Number of vacancies = 3\displaystyle 3
- Constraint: An elector can vote for any number of candidates not exceeding the number of vacancies (i.e., at most 3\displaystyle 3 candidates).

This means the elector can choose to vote for:
1. Exactly 1\displaystyle 1 candidate, OR
2. Exactly 2\displaystyle 2 candidates, OR
3. Exactly 3\displaystyle 3 candidates.

Let us calculate the number of ways for each case:
- **Case 1: Voting for 1 candidate**
Choosing 1\displaystyle 1 candidate out of 5\displaystyle 5:
Ways1=5C1=5\text{Ways}_1 = ^{5}C_{1} = 5
- **Case 2: Voting for 2 candidates**
Choosing 2\displaystyle 2 candidates out of 5\displaystyle 5:
Ways2=5C2=5×42×1=10\text{Ways}_2 = ^{5}C_{2} = \frac{5 \times 4}{2 \times 1} = 10
- **Case 3: Voting for 3 candidates**
Choosing 3\displaystyle 3 candidates out of 5\displaystyle 5:
Ways3=5C3=5C2=10\text{Ways}_3 = ^{5}C_{3} = ^{5}C_{2} = 10
By the addition rule of counting, the total number of ways one can cast their votes is:
Total Ways=Ways1+Ways2+Ways3\text{Total Ways} = \text{Ways}_1 + \text{Ways}_2 + \text{Ways}_3
Total Ways=5+10+10=25\text{Total Ways} = 5 + 10 + 10 = 25

Hence, **Option C** 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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