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12 Dec 2019, 06:18
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Thank you VeritasKarishma,
I think I have got it now. Through my logic I was double counting cases where Top and Bottom would have same pair of colours twice, and excluding certain combinations for the middle faces. Due to this my result was same as the correct answer but had an extremely flawed reasoning.
Thank you again for helping out
I think I have got it now. Through my logic I was double counting cases where Top and Bottom would have same pair of colours twice, and excluding certain combinations for the middle faces. Due to this my result was same as the correct answer but had an extremely flawed reasoning.
Thank you again for helping out
09 May 2021, 06:55
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VeritasKarishma
VeritasPrepKarishma
Why don't we consider the 6 ways in which we can color the top face?
Which face is the top face? All faces are identical. You pick any color and put it on any one side. This can be done in one way only. This is like placing the first person at a round table. All places are identical so place the first one can be put anywhere. Similarly, the first paint can be put on any face of the cube. Now you have a top face (which we have just painted) and a bottom face and 4 identical sides.
Which face is the top face? All faces are identical. You pick any color and put it on any one side. This can be done in one way only. This is like placing the first person at a round table. All places are identical so place the first one can be put anywhere. Similarly, the first paint can be put on any face of the cube. Now you have a top face (which we have just painted) and a bottom face and 4 identical sides.
Responding to a pm:
Quote:
Unfortunately, the way I did this one, I am getting only 15 ways of painting the cube. Wanted your views on this:
As you rightly mentioned, all six faces are identical. So we begin by painting a pair of opposite faces: this pair forms the TOP and BOTTOM of the cube. We can choose the colors by 6C2 (for the TOP and BOTTOM faces). Rest of the 4 sides are identical by all means and the order in which these are painted does not matter reorienting a cube to look like another cube, then the two cubes are not distinct
So total number of ways remain 6C2 only.
PS: The TOP and BOTTOM faces are also exchangeable since the cube can be reoriented (turned upside down). Hence I did not add 2! to my answer.
Note that 6C2 is the number of ways in which you select 2 faces out of 6 DISTINCT faces. The case here is that you have 6 IDENTICAL faces. You pick any one for a colour of your choice in 1 way and call it TOP. Now you automatically have a fixed BOTTOM face. So there is no choosing. You choose a colour for it in 5 ways. As you rightly mentioned, all six faces are identical. So we begin by painting a pair of opposite faces: this pair forms the TOP and BOTTOM of the cube. We can choose the colors by 6C2 (for the TOP and BOTTOM faces). Rest of the 4 sides are identical by all means and the order in which these are painted does not matter reorienting a cube to look like another cube, then the two cubes are not distinct
So total number of ways remain 6C2 only.
PS: The TOP and BOTTOM faces are also exchangeable since the cube can be reoriented (turned upside down). Hence I did not add 2! to my answer.
Next, you have 4 IDENTICAL sides. You pick any 1 in 1 way for a colour of your choice. The rest of the 3 faces are DISTINCT now so you can distribute the 3 colours to them in 3! ways.
So total ways = 5*3! = 30
I understand that, to choose a face we have 1 way since all the faces are identical , but still we have 6 options to color that particular face , then why are we not doing 6*1 ?
10 May 2021, 04:48
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Fin27
VeritasKarishma
VeritasPrepKarishma
Why don't we consider the 6 ways in which we can color the top face?
Which face is the top face? All faces are identical. You pick any color and put it on any one side. This can be done in one way only. This is like placing the first person at a round table. All places are identical so place the first one can be put anywhere. Similarly, the first paint can be put on any face of the cube. Now you have a top face (which we have just painted) and a bottom face and 4 identical sides.
Which face is the top face? All faces are identical. You pick any color and put it on any one side. This can be done in one way only. This is like placing the first person at a round table. All places are identical so place the first one can be put anywhere. Similarly, the first paint can be put on any face of the cube. Now you have a top face (which we have just painted) and a bottom face and 4 identical sides.
Responding to a pm:
Quote:
Unfortunately, the way I did this one, I am getting only 15 ways of painting the cube. Wanted your views on this:
As you rightly mentioned, all six faces are identical. So we begin by painting a pair of opposite faces: this pair forms the TOP and BOTTOM of the cube. We can choose the colors by 6C2 (for the TOP and BOTTOM faces). Rest of the 4 sides are identical by all means and the order in which these are painted does not matter reorienting a cube to look like another cube, then the two cubes are not distinct
So total number of ways remain 6C2 only.
PS: The TOP and BOTTOM faces are also exchangeable since the cube can be reoriented (turned upside down). Hence I did not add 2! to my answer.
Note that 6C2 is the number of ways in which you select 2 faces out of 6 DISTINCT faces. The case here is that you have 6 IDENTICAL faces. You pick any one for a colour of your choice in 1 way and call it TOP. Now you automatically have a fixed BOTTOM face. So there is no choosing. You choose a colour for it in 5 ways. As you rightly mentioned, all six faces are identical. So we begin by painting a pair of opposite faces: this pair forms the TOP and BOTTOM of the cube. We can choose the colors by 6C2 (for the TOP and BOTTOM faces). Rest of the 4 sides are identical by all means and the order in which these are painted does not matter reorienting a cube to look like another cube, then the two cubes are not distinct
So total number of ways remain 6C2 only.
PS: The TOP and BOTTOM faces are also exchangeable since the cube can be reoriented (turned upside down). Hence I did not add 2! to my answer.
Next, you have 4 IDENTICAL sides. You pick any 1 in 1 way for a colour of your choice. The rest of the 3 faces are DISTINCT now so you can distribute the 3 colours to them in 3! ways.
So total ways = 5*3! = 30
I understand that, to choose a face we have 1 way since all the faces are identical , but still we have 6 options to color that particular face , then why are we not doing 6*1 ?
Fin27
Think of circular arrangement. When you place the first person, and make him sit on any seat, do you choose a person? No. You just pick anyone and put him anywhere. After that all seats become distinct. You could start with any person, it won't matter. What matters is how they are placed relative to each other, not in which order you made them sit. Whether A sat first or B sat first is irrelevant. At the end of it, it is only about who is sitting where relative to each other.
Similarly here, initially, all 6 faces are identical. You put any one colour on any face in 1 way. Now your opposite face is distinct but the other 4 faces are still identical.
