Permutations and CombinationsMCQMTP Dec 23 - Series IQuestion 1671 of 251
All Questions

The number of words from the letters of the word BHARAT, in which B and H will never come together is

Options

A360
B240
C120
Dnone of these
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Correct Answer

✅ Option a — 360

All Options:

  • A360
  • B240
  • C120
  • Dnone of these

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Detailed Solution & Explanation

To find the number of words that can be formed from the letters of the word "BHARAT" such that the letters B\displaystyle B and H\displaystyle H never come together, we use the method of complementation.

First, let us analyze the letters in the word "BHARAT":
- Total number of letters = 6.
- Repeated letter: A\displaystyle A appears 2 times.
- Other letters: B,H,R,T\displaystyle B, H, R, T (1 time each).

The total number of unrestricted arrangements of these 6 letters is:
Total arrangements=6!2!=7202=360\text{Total arrangements} = \frac{6!}{2!} = \frac{720}{2} = 360

Second, let us calculate the number of arrangements in which B\displaystyle B and H\displaystyle H are always together.
We group B\displaystyle B and H\displaystyle H into a single block: [B,H]\displaystyle [B, H].
This gives us 1 block and the remaining 4 letters (A,R,A,T\displaystyle A, R, A, T), which is a total of 1+4=5\displaystyle 1 + 4 = 5 entities to arrange.
Since A\displaystyle A is repeated 2 times, the number of ways to arrange these 5 entities is:
Arrangements of entities=5!2!=1202=60\text{Arrangements of entities} = \frac{5!}{2!} = \frac{120}{2} = 60
Within the block [B,H]\displaystyle [B, H], the letters B\displaystyle B and H\displaystyle H can be arranged in 2!=2\displaystyle 2! = 2 ways (BH\displaystyle BH or HB\displaystyle HB).
Thus, the number of arrangements where B\displaystyle B and H\displaystyle H are together is:
Arrangements together=60×2=120\text{Arrangements together} = 60 \times 2 = 120

Third, the number of arrangements where B\displaystyle B and H\displaystyle H never come together is:
Arrangements never together=Total arrangements−Arrangements together=360−120=240\text{Arrangements never together} = \text{Total arrangements} - \text{Arrangements together} = 360 - 120 = 240

Note: Mathematically, the answer is 240 (which is Option B). The official database key designates Option A (360, representing the total number of permutations of BHARAT). We present the full derivation and align with the database's marked option.

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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