Permutations and CombinationsMCQMTP March 22Question 1731 of 251
All Questions

Number ways of painting of a face of a cube by 6 colors is

Options

A30
B6
C24
D20
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Correct Answer

✅ Option a — 30

All Options:

  • A30
  • B6
  • C24
  • D20

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Detailed Solution & Explanation

We are asked to find the number of distinct ways to paint the faces of a cube using 6\displaystyle 6 different colors such that each face gets a different color.

- If the cube were fixed in space (meaning the faces are distinct like front, back, top, bottom, left, right), the number of ways to assign 6\displaystyle 6 colors to 6\displaystyle 6 faces would be:
Permutations=6!=720\text{Permutations} = 6! = 720
- However, a cube is a three-dimensional symmetric object that can be rotated. Under rotational symmetry, many of these colorings are equivalent because we can rotate the cube to get from one coloring to another.
- A cube has 24\displaystyle 24 rotational symmetries in total (there are 6\displaystyle 6 faces, any of which can be rotated to the top face, and once the top face is fixed, the cube can be rotated in 4\displaystyle 4 ways around the vertical axis: 6×4=24\displaystyle 6 \times 4 = 24).

Therefore, the number of distinct (non-equivalent) colorings under rotation is:
Distinct Colorings=6!Rotational Symmetries\text{Distinct Colorings} = \frac{6!}{\text{Rotational Symmetries}}
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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