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805+ (Hard)|   Combinations|      
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megha10
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You have a six-sided cube and six cans of paint, each a diff 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 :)
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You have a six-sided cube and six cans of paint, each a diff 09 May 2021, 06:55
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VeritasKarishma
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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.

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.
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
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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 ?
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You have a six-sided cube and six cans of paint, each a diff 10 May 2021, 04:48
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Fin27
VeritasKarishma
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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.

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.
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 ?
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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.
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You have a six-sided cube and six cans of paint, each a diff 10 May 2021, 05:10
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Bunuel VeritasKarishma

Why (6-1)! is not the applicable here?
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You have a six-sided cube and six cans of paint, each a diff 10 May 2021, 05:29
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VeritasKarishma

So basically if iam doing 1*6C1, Iam assuming an order in which first color has to be chosen , which is not required, am I right?
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You have a six-sided cube and six cans of paint, each a diff 11 May 2021, 02:14
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Bunuel VeritasKarishma

Why (6-1)! is not the applicable here?
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Because after you select a face and put a colour on it, the face opposite to it becomes distinct.
But all the other 4 faces on the SIDES are still identical. Look at a cube to understand this. If you have colours on top and bottom faces and no colour or same colour on all side faces, all side faces are identical.

Which one you start with doesn't matter. So you again place a colour on any one of the 4 faces in 1 way. Now the other 3 faces are distinct - one is opposite to the coloured face, one is on the right and one is on the left.

In ciurlar arrangements, the moment you place the first person, all other seats become distinct.
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You have a six-sided cube and six cans of paint, each a diff 11 May 2021, 02:22
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Fin27
VeritasKarishma

So basically if iam doing 1*6C1, Iam assuming an order in which first color has to be chosen , which is not required, am I right?
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Yes. In that case you are choosing the order in which each colour is put even though it is irrelevant.

Think about it: Say you use 6C1 and pick the colour Blue as your first colour and put it on a face. Next, you do 5C1 and put green on the opposite face and then put rest 4 colours on other 4 sides.

Then there will be a case in 6C1 in which you pick green as your first colour and put it on a face. Now, your 5C1 includes the case of blue on the opposite face. Again you put the rest of the 4 colours in the same way on the other 4 sides.

The two cubes you will obtain are identical but you counted them as 2 cases when you used 6C1.
Instead, pick any colour and put it on a face. Now you do 5C1 to pick one of 5 colours on the opposite face. So this will always give you a distinct pair of colours on these opposite faces.
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You have a six-sided cube and six cans of paint, each a diff 21 Nov 2022, 09:02
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chetan86
You have a six-sided cube and six cans of paint, each a different color. You may not mix colors of paint. How many distinct ways can you paint the cube using a different color for each side? (If you can reorient a cube to look like another cube, then the two cubes are not distinct.)

(A) 24
(B) 30
(C) 48
(D) 60
(E) 120
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Let's assume that the cube is actually a dice with six side numbered 1-6. Now you have six colours. Let' say you paint colour cube in this way

1R. 6W. (1 is opposite 6)
2B. 4G. (2 is opposite 4)
3Y. 5P. (3 is opposite 5)

Now if we have another cube where
6R 1W
2G 4B
3P 5Y
Then the cube is identical.

But the we can swap colour combination between 3 dimensions in 3! Ways.

At the same time Red can be combined with 5 other colours on opposite to create different colour combination. Example 1R 6B
Same for other side
Hence total number of distinct ways = 3!*5 = 30 ways

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You have a six-sided cube and six cans of paint, each a diff 22 Apr 2026, 13:35
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