Number of distinct outcomes: dual-colored cube

[Gascoigne]

If each side of a cube can be either red or blue, how many distinct possibilities for a single cube are there in total?

2 for all R or all B
12 for one R or one B
30 for two R or two B
40 for three R or three B
30 for four R or four B
12 for five R or five B
= 105

Is there a better way to do this than to calculate the combinations with n!/r!(n-r)! for each number of altered colors?

 

[chivox]

If each side of a cube can be either red or blue, how many distinct possibilities for a single cube are there in total?

2 for all R or all B
12 for one R or one B
30 for two R or two B >>> Aren't these the same cubes
40 for three R or three B
30 for four R or four B >>> as these?? If you have 4 R's, how many B's do you have?
12 for five R or five B
= 105

Is there a better way to do this than to calculate the combinations with n!/r!(n-r)! for each number of altered colors?

If the sides of the cube are numbered in a way that makes "Side 1" of the cube different from all the other sides, each of six sides has two choices. That's all the information you need to do this.

Here's a simpler example: How many ways are there to arrange R and B in three slots? Let's enumerate them.

RRR
RRB
RBR
BRR
RBB
BBR
BRB
BBB

There are eight, says the owl. In the cube problem, we're not finding how many two-color combinations there are out of six colors here (that would be a combinations solution, 6C2), but rather, we're putting colors on a cube.

 

[Gascoigne]

"How many distinct patterns are possible, including all six sides of the cube in each pattern?"

For this problem, aren't there 6 ways the cube can be 1 R and 5 B?
RBBBBB
BRBBBB
BBRBBB
BBBRBB
BBBBRB
BBBBBR

Are these not distinct because they're the same pattern, just on different sides?

 

[soroban]

Hello, Gascoigne!

If each side of a cube can be either red or blue,
how many distinct possibilities for a single cube are there in total?

If we are not allowed to rotate or reorient the cube,
. . then there are: .\(\displaystyle 2^6 \,=\,64\) different colorings.


Problems of this type usually allow rotation of the cube,
. . drastically reducing the number of colorings.

          * - - - - - *
         /           /|
        /     T     / |
       /           /  |
      * - - - - - *   |
      |           | R *
      |           |  /
      |     F     | /
      |           |/
      * - - - - - *

We can label some of the faces: .Front, Top, Right, and Left.


0 red, 6 blue: 1 way
0 blue, 6 red: 1 way
. . There is only one way to paint a cube in one color.


1 red, 5 blue: 1 way
1 blue, 5 red: 1 way
. . That one red can be any of the six faces; it doesn't matter.


2 red, 4 blue: 2 ways
2 blue, 4 red: 2 ways
. . The two red faces can be adjacent (T and R) or opposite (L and R) ... two choices.


3 red, 3 blue: 2 ways
. . The three red faces can be adjacent (T, F, R)
. . . or form a row (L, F, R) ... two choices.


I count ten distinct colorings of the cube.

 

[Gascoigne]

Hello, Gascoigne!

If each side of a cube can be either red or blue,
how many distinct possibilities for a single cube are there in total?

If we are not allowed to rotate or reorient the cube,
. . then there are: .\(\displaystyle 2^6 \,=\,64\) different colorings.


Problems of this type usually allow rotation of the cube,
. . drastically reducing the number of colorings.

          * - - - - - *
         /           /|
        /     T     / |
       /           /  |
      * - - - - - *   |
      |           | R *
      |           |  /
      |     F     | /
      |           |/
      * - - - - - *

We can label some of the faces: .Front, Top, Right, and Left.


0 red, 6 blue: 1 way
0 blue, 6 red: 1 way
. . There is only one way to paint a cube in one color.


1 red, 5 blue: 1 way
1 blue, 5 red: 1 way
. . That one red can be any of the six faces; it doesn't matter.


2 red, 4 blue: 2 ways
2 blue, 4 red: 2 ways
. . The two red faces can be adjacent (T and R) or opposite (L and R) ... two choices.


3 red, 3 blue: 2 ways
. . The three red faces can be adjacent (T, F, R)
. . . or form a row (L, F, R) ... two choices.


I count ten distinct colorings of the cube.

Well, the problem was derived from someone else's - the instructor changed diagonals to colors. I thought the same methodology would be used regardless:
Suppose a diagonal is drawn across each side of a cube from one corner to the other. How many distinct patterns are possible, considering all possible orientations of the diagonals, including all six sides of the cube in each pattern?

Is this a different problem than my colors one, due to positioning?

 

[chivox]

"How many distinct patterns are possible, including all six sides of the cube in each pattern?"

For this problem, aren't there 6 ways the cube can be 1 R and 5 B?
RBBBBB
BRBBBB
BBRBBB
BBBRBB
BBBBRB
BBBBBR

Are these not distinct because they're the same pattern, just on different sides?

Yes, they're distinct. However, my point was that you counted those six ways twice: once when you said "1 R or 1 B" and once when you said "5 R or 5 B". Half the chart you made was double counted, which since the "3 R or 3 B" case also contained duplicates of each cube, means you would have 64 distinct cubes, which is just 2^6.