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# You have a six-sided cube and six cans of paint, each a diff

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You have a six-sided cube and six cans of paint, each a diff  [#permalink]

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09 Sep 2013, 06:45
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Question Stats:

30% (01:56) correct 70% (01:56) wrong based on 169 sessions

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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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Re: You have a six-sided cube and six cans of paint, each a diff  [#permalink]

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10 Sep 2013, 06:16
9
8
chetan86 wrote:
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

Paint one of the faces red and make it the top face.
5 options for the bottom face.
Now, four side faces can be painted in (4-1)! = 3! = 6 ways (circular arrangements of 4 colors).

Total = 5*6 = 30.

Similar question to practice: a-cube-marked-1-2-3-4-5-and-6-on-its-six-faces-three-89198.html
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Re: You have a six-sided cube and six cans of paint, each a diff  [#permalink]

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28 May 2014, 03:41
5
jlgdr wrote:
Bunuel wrote:
chetan86 wrote:
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

Paint one of the faces red and make it the top face.
5 options for the bottom face.
Now, four side faces can be painted in (4-1)! = 3! = 6 ways (circular arrangements of 4 colors).

Total = 5*6 = 30.

Similar question to practice: http://gmatclub.com/forum/a-cube-marked ... 89198.html

I really didn't get how you used circular arrangements here.

We have six different colors and 6 different sides.

How many ways can we paint the cube with each side a different color?

Shouldn't it be 6! ?

Great problem btw

Cheers
J

Painting a cube with six different colors is more complicated than arranging six distinct things in a row.

If you have six distinct colors, say ROYGBP, you could place them in the slots below in 6! ways. Why? Any of the 6 colors could go in the first slot, any of the five remaining could go in the second, and so on. So the total is 6*5*4*3*2*1=6!
_ _ _ _ _ _

Now the cube is a combination of slots and a circular arrangement. Here is my sophisticated diagram for that:

_
_ _ _ _
_

The four slots between the top and bottom 'faces' actually wrap around the whole cube. So, let's say for the four 'slots' you choose ROYG.

ROYG =GROY = YGRO = OYGR = ROYG

All of these arrangements are the same because essentially each one is just a rotation of the cube by 90 degrees, not a different paint job.

So we treat the four 'slots' in the middle the same way we would a circular table. Hence the above solutions: you can choose any color for the top face, you can choose one of the five remaining colors for the bottom face (5 ways), and since the four middle faces are 'in a circle' they can be arranged (4-1)!=3! ways. So the total is 5*3!

The tricky part is that we don't count the ways in which we can choose the color of the first face, since every color is going to be chosen anyway. Essentially, you are finding the ways you can paint the other sides relative to one of painted sides. Otherwise you are including in your total the number of different ways you can look at the cube (which don't constitute a new paint job).

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Re: You have a six-sided cube and six cans of paint, each a diff  [#permalink]

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10 Sep 2013, 07:56
Bunuel wrote:
chetan86 wrote:
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

Paint one of the faces red and make it the top face.
5 options for the bottom face.
Now, four side faces can be painted in (4-1)! = 3! = 6 ways (circular arrangements of 4 colors).

Total = 5*6 = 30.

Similar question to practice: a-cube-marked-1-2-3-4-5-and-6-on-its-six-faces-three-89198.html

Thanks Bunues, great explanation!!
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Re: You have a six-sided cube and six cans of paint, each a diff  [#permalink]

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27 Dec 2013, 20:12
Bunuel wrote:
chetan86 wrote:
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

Paint one of the faces red and make it the top face.
5 options for the bottom face.
Now, four side faces can be painted in (4-1)! = 3! = 6 ways (circular arrangements of 4 colors).

Total = 5*6 = 30.

Similar question to practice:

Can you give more detail on how you determined what method to use to solve this problem? I have never heard of the "circular arrangement" technique you used.
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Posts: 58402
Re: You have a six-sided cube and six cans of paint, each a diff  [#permalink]

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28 Dec 2013, 03:56
5
4
sjmarinov wrote:
Bunuel wrote:
chetan86 wrote:
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

Paint one of the faces red and make it the top face.
5 options for the bottom face.
Now, four side faces can be painted in (4-1)! = 3! = 6 ways (circular arrangements of 4 colors).

Total = 5*6 = 30.

Similar question to practice:

Can you give more detail on how you determined what method to use to solve this problem? I have never heard of the "circular arrangement" technique you used.

The number of arrangements of n distinct objects in a row is given by $$n!$$.
The number of arrangements of n distinct objects in a circle is given by $$(n-1)!$$.

From Gmat Club Math Book (combinatorics chapter):
"The difference between placement in a row and that in a circle is following: if we shift all object by one position, we will get different arrangement in a row but the same relative arrangement in a circle. So, for the number of circular arrangements of n objects we have:

$$R = \frac{n!}{n} = (n-1)!$$"

seven-men-and-seven-women-have-to-sit-around-a-circular-92402.html
a-group-of-four-women-and-three-men-have-tickets-for-seven-a-88604.html
the-number-of-ways-in-which-5-men-and-6-women-can-be-seated-94915.html
4-couples-are-seating-at-a-round-tables-how-many-ways-can-131048.html
at-a-party-5-people-are-to-be-seated-around-a-circular-104101.html
seven-family-members-are-seated-around-their-circular-dinner-102184.html
seven-men-and-seven-women-have-to-sit-around-a-circular-11473.html
a-group-of-8-friends-sit-together-in-a-circle-alice-betty-106928.html
seven-men-and-five-women-have-to-sit-around-a-circular-table-98185.html
a-group-of-8-friends-sit-together-in-a-circle-alice-betty-106928.html
find-the-number-of-ways-in-which-four-men-two-women-and-a-106919.html
gmat-club-monday-giveaway-155157.html (700+)

Theory on Combinations: math-combinatorics-87345.html

DS questions on Combinations: search.php?search_id=tag&tag_id=31
PS questions on Combinations: search.php?search_id=tag&tag_id=52

Tough and tricky questions on Combinations: hardest-area-questions-probability-and-combinations-101361.html

Hope this helps.
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Re: You have a six-sided cube and six cans of paint, each a diff  [#permalink]

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28 Dec 2013, 21:30
This helps a great deal! Thank you Bunuel!
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Re: You have a six-sided cube and six cans of paint, each a diff  [#permalink]

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27 May 2014, 06:19
Bunuel wrote:
chetan86 wrote:
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

Paint one of the faces red and make it the top face.
5 options for the bottom face.
Now, four side faces can be painted in (4-1)! = 3! = 6 ways (circular arrangements of 4 colors).

Total = 5*6 = 30.

Similar question to practice: a-cube-marked-1-2-3-4-5-and-6-on-its-six-faces-three-89198.html

I really didn't get how you used circular arrangements here.

We have six different colors and 6 different sides.

How many ways can we paint the cube with each side a different color?

Shouldn't it be 6! ?

Great problem btw

Cheers
J
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You have a six-sided cube and six cans of paint, each a diff  [#permalink]

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15 Oct 2014, 22:32
Bunuel wrote:
chetan86 wrote:
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

Paint one of the faces red and make it the top face.
5 options for the bottom face.
Now, four side faces can be painted in (4-1)! = 3! = 6 ways (circular arrangements of 4 colors).

Total = 5*6 = 30.

@Bunuel

, why haven't the ways to paint the first (top)face have been counted, i.e 6C1 ways to choose from amongst the 6 diffferent paints?? And what does the wording provided in the question in the parentheses entails?
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Re: You have a six-sided cube and six cans of paint, each a diff  [#permalink]

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15 Jan 2015, 22:57
Bunuel wrote:
chetan86 wrote:
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

Paint one of the faces red and make it the top face.
5 options for the bottom face.
Now, four side faces can be painted in (4-1)! = 3! = 6 ways (circular arrangements of 4 colors).

Total = 5*6 = 30.

Similar question to practice: a-cube-marked-1-2-3-4-5-and-6-on-its-six-faces-three-89198.html

Hi Bunuel,
Need your help on this. I could understand the circular and bottom face things. that's 5*(4-1)!. But I am not able to understand that top part. If initially no sides are painted, then we could chose 1 among 6 paints right? so shouldn't we multiply 5*(4-1)! with 6, as we have six choices initially?
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Re: You have a six-sided cube and six cans of paint, each a diff  [#permalink]

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15 Jan 2015, 23:29
1
Vinitkhicha1111 wrote:
Bunuel wrote:
chetan86 wrote:
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

Paint one of the faces red and make it the top face.
5 options for the bottom face.
Now, four side faces can be painted in (4-1)! = 3! = 6 ways (circular arrangements of 4 colors).

Total = 5*6 = 30.

@Bunuel

, why haven't the ways to paint the first (top)face have been counted, i.e 6C1 ways to choose from amongst the 6 diffferent paints?? And what does the wording provided in the question in the parentheses entails?

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.
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Re: You have a six-sided cube and six cans of paint, each a diff  [#permalink]

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23 Jun 2017, 08:00
@Bunuel

What about this one? Is this a GMAT question? or this too is out of scope? I found a lot of difficulty solving this one..and got it wrong the first time.

Sent from my Pixel XL using GMAT Club Forum mobile app
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Re: You have a six-sided cube and six cans of paint, each a diff  [#permalink]

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23 Jun 2017, 08:22
ShashankDave wrote:
Bunuel

What about this one? Is this a GMAT question? or this too is out of scope? I found a lot of difficulty solving this one..and got it wrong the first time.

Sent from my Pixel XL using GMAT Club Forum mobile app

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No, this question is fine.
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Re: You have a six-sided cube and six cans of paint, each a diff  [#permalink]

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23 Jun 2017, 11:19
Bunuel wrote:
ShashankDave wrote:
Bunuel

What about this one? Is this a GMAT question? or this too is out of scope? I found a lot of difficulty solving this one..and got it wrong the first time.

Sent from my Pixel XL using GMAT Club Forum mobile app

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No, this question is fine.

Okay. Please suggest somewhere to read about these types of problems..and can you also explain any other method to approach this question? Combinatorics is kind of my strength, I don't want to miss questions such as these on the GMAT. Please help me.
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Re: You have a six-sided cube and six cans of paint, each a diff  [#permalink]

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23 Jun 2017, 11:35
ShashankDave wrote:
Bunuel wrote:
ShashankDave wrote:
Bunuel

What about this one? Is this a GMAT question? or this too is out of scope? I found a lot of difficulty solving this one..and got it wrong the first time.

Sent from my Pixel XL using GMAT Club Forum mobile app

___________________
No, this question is fine.

Okay. Please suggest somewhere to read about these types of problems..and can you also explain any other method to approach this question? Combinatorics is kind of my strength, I don't want to miss questions such as these on the GMAT. Please help me.

This question does not represent any particular/special type of combinatorics questions - we use circular permutation there and that's it. Links to circular permutation questions are given above. Don't know what else to suggest. Hopefully more practice should help.
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Re: You have a six-sided cube and six cans of paint, each a diff  [#permalink]

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12 Jul 2017, 04:51
VeritasPrepKarishma wrote:
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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Re: You have a six-sided cube and six cans of paint, each a diff  [#permalink]

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22 Aug 2018, 10:52
While painting the top face..why don't we count the number of ways of choosing a colour.

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Re: You have a six-sided cube and six cans of paint, each a diff  [#permalink]

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23 Aug 2018, 10:18
MithilaGauri wrote:
While painting the top face..why don't we count the number of ways of choosing a colour.

Posted from my mobile device

I have addressed it here: https://gmatclub.com/forum/you-have-a-s ... l#p1469980
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Re: You have a six-sided cube and six cans of paint, each a diff  [#permalink]

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Re: You have a six-sided cube and six cans of paint, each a diff   [#permalink] 27 Aug 2019, 09:22
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