Permutations and CombinationsMCQMTP June 24 Series IIIQuestion 1751 of 251
All Questions

The number of ways of painting the six faces of a cube with six different given colours is

Options

A1\displaystyle 1
B720\displaystyle 720
C30\displaystyle 30
D15\displaystyle 15
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Correct Answer

✅ Option c — 30\displaystyle 30

All Options:

  • A1\displaystyle 1
  • B720\displaystyle 720
  • C30\displaystyle 30
  • D15\displaystyle 15

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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 using 6\displaystyle 6 different colors under rotational symmetry.

- If the cube had fixed, distinct 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
- Because the cube is free to rotate in three dimensions, many of these colorings are equivalent. A cube has 24\displaystyle 24 rotational symmetries:
- Any of the 6\displaystyle 6 faces can be placed at the top (6\displaystyle 6 choices).
- For each top face, the cube can be rotated in 4\displaystyle 4 orientations 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 under rotation:
Distinct Colorings=6!24=72024=30\text{Distinct Colorings} = \frac{6!}{24} = \frac{720}{24} = 30

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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