Permutations and CombinationsMCQMTP Dec 22 - Series IQuestion 1734 of 251
All Questions

The number of ways of painting the faces of a cube by 6 different colors is

Options

A30
B36
C24
D1
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Correct Answer

✅ Option a — 30

All Options:

  • A30
  • B36
  • C24
  • D1

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Detailed Solution & Explanation

We want to find the number of distinct ways to color the 6\displaystyle 6 faces of a cube with 6\displaystyle 6 different colors under rotational symmetry.

If the cube had fixed, identifiable faces in space, the number of ways to assign 6\displaystyle 6 colors to the 6\displaystyle 6 faces would be:
Total Permutations=6!=720\text{Total Permutations} = 6! = 720
However, because the cube is free to rotate in three dimensions, many of these colorings are identical under rotation. A cube has 24\displaystyle 24 rotational symmetries:
- Any of the 6\displaystyle 6 faces can be chosen to be the top face (6\displaystyle 6 choices).
- For each choice, the cube can be rotated in 4\displaystyle 4 ways around the vertical axis (4\displaystyle 4 choices).
- Total rotational symmetries = 6×4=24\displaystyle 6 \times 4 = 24.

Dividing the total permutations by the number of equivalent orientations:
Distinct Colorings=6!24=72024=30\text{Distinct Colorings} = \frac{6!}{24} = \frac{720}{24} = 30

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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