Permutations and CombinationsMCQMTP June 24 Series IQuestion 1677 of 251
All Questions

Find the number of arrangements in which the letters of the word 'MONDAY' be arranged so that the words thus formed begin with 'M' and do not end with 'N'.

Options

A720
B120
C96
DNone of these
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Correct Answer

✅ Option c — 96

All Options:

  • A720
  • B120
  • C96
  • DNone of these

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Detailed Solution & Explanation

We want to find the number of permutations of the 6-letter word "MONDAY" such that each arrangement:
1. Begins with the letter 'M'.
2. Does not end with the letter 'N'.

The word "MONDAY" consists of 6 distinct letters: {M,O,N,D,A,Y}\displaystyle \{M, O, N, D, A, Y\}.
Let us denote the positions in our 6-letter word as 1,2,3,4,5,6\displaystyle 1, 2, 3, 4, 5, 6.

- **Position 1:** Must be filled by the letter 'M'. Since there is only one 'M', there is exactly 1 way to fill this position.
- **Remaining Positions:** We have 5 positions left (2,3,4,5,6\displaystyle 2, 3, 4, 5, 6) and 5 remaining letters {O,N,D,A,Y}\displaystyle \{O, N, D, A, Y\}.

- **Position 6 (Last Position):** Cannot contain the letter 'N'. Thus, it must be filled by one of the 4 letters from the set {O,D,A,Y}\displaystyle \{O, D, A, Y\}.
Ways to fill Position 6=4 ways\text{Ways to fill Position 6} = 4 \text{ ways}

- **Positions 2, 3, 4, 5:** Once Position 6 is filled, we have 4 remaining letters (which includes 'N' if it was not chosen for Position 6) to be arranged in the 4 remaining positions. The number of ways to arrange these 4 letters is:
4!=24 ways4! = 24 \text{ ways}

By the fundamental multiplication principle of counting, the total number of valid arrangements is:
Total arrangements=1×4×4!=4×24=96\text{Total arrangements} = 1 \times 4 \times 4! = 4 \times 24 = 96

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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