While finding the number of ways to paint a cube, why didnâ€™t we consider a factor of 6 ways for selecting the first colour (red in the given example)?

We are assuming a cube freely floating in space (i.e. i can turn/twist it in any direction). So when unpainted, all the faces will be effectively identical. So since every colour has to appear once, I just arbitrarily chose a colour and a position for it and then decided where to place the other colours relative to it. You could have started with any colour, it would make no difference.

But sir while fixing 1 side with a color , what is wrong with the following :-
like first choosing a side from 6 identical sides= 1 way
and then coloring it with any of the 6 colors =6c1 ways
so shouldn’t it be 6c1*1 ?

Until you have painted one colour, the sides don’t become distinct. Everything will be measured wrt to the first painted side.

regards
J

In the cube case,after painting first two opposite sides,the remaining four sides can be viewed as four places in a circle.So they can be arranged in 3! ways.So total no of ways is 5*3!=30 ways.

I think for the cuboid the ways of painting should be 360

Hi Sir,

While finding the number of ways to paint a cube, why didnâ€™t we consider a factor of 6 ways for selecting the first colour (red in the given example)?

We are assuming a cube freely floating in space (i.e. i can turn/twist it in any direction). So when unpainted, all the faces will be effectively identical. So since every colour has to appear once, I just arbitrarily chose a colour and a position for it and then decided where to place the other colours relative to it. You could have started with any colour, it would make no difference.

regards

J

But sir while fixing 1 side with a color , what is wrong with the following :-

like first choosing a side from 6 identical sides= 1 way

and then coloring it with any of the 6 colors =6c1 ways

so shouldn’t it be 6c1*1 ?

Until you have painted one colour, the sides don’t become distinct. Everything will be measured wrt to the first painted side.

regards

J

In the cube case,after painting first two opposite sides,the remaining four sides can be viewed as four places in a circle.So they can be arranged in 3! ways.So total no of ways is 5*3!=30 ways.

why is 5 multiplied?