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Permutation or groups!: TIFR 2019 GS Part A, Problem 18

Understand the problem

Consider the different ways to colour a cube with 6 given colours such that each face will be given a single colour and all the six colours will be used. Define two such colourings to be equivalent if one will get from another just by rotation. Then what would be the number of inequivalent colourings?

Source of the problem

TIFR 2019 GS Part A, Problem 18

Topic
Combinatorics
Difficulty Level
Moderate

Start with hints

Do you really need a hint? Try it first!

For example, first let us choose the colours on the two opposite sides in ^6C_2 ways, then another two opposite sides in ^4C_2 ways, and the leftover sides in ^2C_2 ways. So, in total, we get 15 \times 6=90, but think about over counting.
you’re choosing three pairs of oppositely placed colours, but you are overcounting because the same division into three pairs can be chosen in
6 different orders, being counted erroneously as distinct by you while. At the same time, you consider one colouring and its mirror image non-distinct (swapping two opposite colours is erroneously seen as the same choice). This gives a total overcounting by a factor of \frac62=3.
Another approach: There are 6!=720 different ways of colouring without taking rotations into consideration. There are 24 different rotations, and for any colouring, no two rotations give identical result. Therefore these 720 colourings may be divided into groups of 24 essentially equal colourings. There must be 30 such groups.
Another approach: In a circular permutation, arrangements which may be obtained from one another by rotating the cube are considered identical. Place a colour on the bottom. It does not matter which. The top colour may be chosen in five ways. Place another colour facing you. Again, it does not matter which. The remaining three colours may be arranged relative to the colour facing you in 3! ways as we proceed clockwise around the cube. Hence, there are 5 \cdot 3! = 30 admissible colourings.

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