Permutations and CombinationsMCQPYQ Jan. 21Question 1623 of 251
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'n\displaystyle n' locks and 'n\displaystyle n' corresponding keys are available, but the actual combination is not known. The maximum number of trials that are needed to assign the keys to the corresponding locks is:

Options

A(n−1)C2\displaystyle (n-1)C_2
B(n+1)C2\displaystyle (n+1)C_2
C∑k=1n(k−1)\displaystyle \sum_{k=1}^{n} (k-1)
D∑k=1nk\displaystyle \sum_{k=1}^{n} k
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Correct Answer

✅ Option c — ∑k=1n(k−1)\displaystyle \sum_{k=1}^{n} (k-1)

All Options:

  • A(n−1)C2\displaystyle (n-1)C_2
  • B(n+1)C2\displaystyle (n+1)C_2
  • C∑k=1n(k−1)\displaystyle \sum_{k=1}^{n} (k-1)
  • D∑k=1nk\displaystyle \sum_{k=1}^{n} k

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Detailed Solution & Explanation

Let us analyze the puzzle using the worst-case scenario (maximum trials) for each lock:
- For the 1st lock: We have n\displaystyle n keys. In the worst case, we try n−1\displaystyle n-1 keys. If none of these open the lock, the remaining 1 key must be the correct one, so we do not need a trial for it. Thus, maximum trials = n−1\displaystyle n-1.
- For the 2nd lock: We have n−1\displaystyle n-1 keys and n−1\displaystyle n-1 locks remaining. In the worst case, we try n−2\displaystyle n-2 keys. Maximum trials = n−2\displaystyle n-2.
- Repeating this, for the (n−1)\displaystyle (n-1)-th lock: We have 2 keys and 2 locks. In the worst case, we try 1 key. Maximum trials = 1\displaystyle 1.
- For the last lock: We have 1 key and 1 lock remaining. This requires 0 trials.

The maximum total number of trials is the sum of these trials:
textTotaltrials=(n−1)+(n−2)+dots+1+0\\text{Total trials} = (n-1) + (n-2) + \\dots + 1 + 0
Using summation notation, this can be written as:
sumk=1n(k−1)\\sum_{k=1}^{n} (k-1)
This matches Option C.
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.

Key Concepts to Understand

Combination

A selection of objects where order does not matter.

Related Comparison Tables

Permutations vs Combinations: The Quantitative Aptitude Guide

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