Given six different colours and a cube.
1. In how many ways two opposite faces of the cube can
be coloured with two different colours?
2. In how many ways four faces of the cube can be
coloured with four different colours of which two
faces are opposite and other two faces are also
opposite?
3. In how many ways all six faces of the cube can be
coloured with six different colours?
Colors 1,2,3,4,5,6
Sides 1 bottom, 2 top, 3 left, 4 right, 5 back, 6 front
1. C(6,2)=6*5/2=15
[[1,2],[1,3],[1,4],[1,5],[1,6],[2,3],[2,4],[2,5],[2,6],
[3,4],[3,5],[3,6],[4,5],[4,6],[5,6]]
2. 15*4*3=180
Flip and rotate.
Without duplicates there are
135 different ways.
[[1,2,3,4],[1,2,3,5],[1,2,3,6],[1,2,4,3],[1,2,4,5],
[1,2,4,6],[1,2,5,3],[1,2,5,4],[1,2,5,6],[1,2,6,3],
[1,2,6,4],[1,2,6,5],[1,3,2,4],[1,3,2,5],[1,3,2,6],
[1,3,4,2],[1,3,4,5],[1,3,4,6],[1,3,5,2],[1,3,5,4],
[1,3,5,6],[1,3,6,2],[1,3,6,4],[1,3,6,5],[1,4,2,3],
[1,4,2,5],[1,4,2,6],[1,4,3,2],[1,4,3,5],[1,4,3,6],
[1,4,5,2],[1,4,5,3],[1,4,5,6],[1,4,6,2],[1,4,6,3],
[1,4,6,5],[1,5,2,3],[1,5,2,4],[1,5,2,6],[1,5,3,2],
[1,5,3,4],[1,5,3,6],[1,5,4,2],[1,5,4,3],[1,5,4,6],
[1,5,6,2],[1,5,6,3],[1,5,6,4],[1,6,2,3],[1,6,2,4],
[1,6,2,5],[1,6,3,2],[1,6,3,4],[1,6,3,5],[1,6,4,2],
[1,6,4,3],[1,6,4,5],[1,6,5,2],[1,6,5,3],[1,6,5,4],
[2,3,1,4],[2,3,1,5],[2,3,1,6],[2,3,4,5],[2,3,4,6],
[2,3,5,4],[2,3,5,6],[2,3,6,4],[2,3,6,5],[2,4,1,3],
[2,4,1,5],[2,4,1,6],[2,4,3,5],[2,4,3,6],[2,4,5,3],
[2,4,5,6],[2,4,6,3],[2,4,6,5],[2,5,1,3],[2,5,1,4],
[2,5,1,6],[2,5,3,4],[2,5,3,6],[2,5,4,3],[2,5,4,6],
[2,5,6,3],[2,5,6,4],[2,6,1,3],[2,6,1,4],[2,6,1,5],
[2,6,3,4],[2,6,3,5],[2,6,4,3],[2,6,4,5],[2,6,5,3],
[2,6,5,4],[3,4,1,2],[3,4,1,5],[3,4,1,6],[3,4,2,5],
[3,4,2,6],[3,4,5,6],[3,4,6,5],[3,5,1,2],[3,5,1,4],
[3,5,1,6],[3,5,2,4],[3,5,2,6],[3,5,4,6],[3,5,6,4],
[3,6,1,2],[3,6,1,4],[3,6,1,5],[3,6,2,4],[3,6,2,5],
[3,6,4,5],[3,6,5,4],[4,5,1,2],[4,5,1,3],[4,5,1,6],
[4,5,2,3],[4,5,2,6],[4,5,3,6],[4,6,1,2],[4,6,1,3],
[4,6,1,5],[4,6,2,3],[4,6,2,5],[4,6,3,5],[5,6,1,2],
[5,6,1,3],[5,6,1,4],[5,6,2,3],[5,6,2,4],[5,6,3,4]]
3. 135*2=270
Flip and rotate.
Without duplicates there are
210 different ways.
[[1,2,3,4,5,6],[1,2,3,4,6,5],[1,2,3,5,4,6],[1,2,3,5,6,4],
[1,2,3,6,4,5],[1,2,3,6,5,4],[1,2,4,3,5,6],[1,2,4,3,6,5],
[1,2,4,5,3,6],[1,2,4,5,6,3],[1,2,4,6,3,5],[1,2,4,6,5,3],
[1,2,5,3,4,6],[1,2,5,3,6,4],[1,2,5,4,3,6],[1,2,5,4,6,3],
[1,2,5,6,3,4],[1,2,5,6,4,3],[1,2,6,3,4,5],[1,2,6,3,5,4],
[1,2,6,4,3,5],[1,2,6,4,5,3],[1,2,6,5,3,4],[1,2,6,5,4,3],
[1,3,2,4,5,6],[1,3,2,4,6,5],[1,3,2,5,4,6],[1,3,2,5,6,4],
[1,3,2,6,4,5],[1,3,2,6,5,4],[1,3,4,2,5,6],[1,3,4,2,6,5],
[1,3,4,5,2,6],[1,3,4,5,6,2],[1,3,4,6,2,5],[1,3,4,6,5,2],
[1,3,5,2,4,6],[1,3,5,2,6,4],[1,3,5,4,2,6],[1,3,5,4,6,2],
[1,3,5,6,2,4],[1,3,5,6,4,2],[1,3,6,2,4,5],[1,3,6,2,5,4],
[1,3,6,4,2,5],[1,3,6,4,5,2],[1,3,6,5,2,4],[1,3,6,5,4,2],
[1,4,2,3,5,6],[1,4,2,3,6,5],[1,4,2,5,3,6],[1,4,2,5,6,3],
[1,4,2,6,3,5],[1,4,2,6,5,3],[1,4,3,2,5,6],[1,4,3,2,6,5],
[1,4,3,5,2,6],[1,4,3,5,6,2],[1,4,3,6,2,5],[1,4,3,6,5,2],
[1,4,5,2,3,6],[1,4,5,2,6,3],[1,4,5,3,2,6],[1,4,5,3,6,2],
[1,4,5,6,2,3],[1,4,5,6,3,2],[1,4,6,2,3,5],[1,4,6,2,5,3],
[1,4,6,3,2,5],[1,4,6,3,5,2],[1,4,6,5,2,3],[1,4,6,5,3,2],
[1,5,2,3,4,6],[1,5,2,3,6,4],[1,5,2,4,3,6],[1,5,2,4,6,3],
[1,5,2,6,3,4],[1,5,2,6,4,3],[1,5,3,2,4,6],[1,5,3,2,6,4],
[1,5,3,4,2,6],[1,5,3,4,6,2],[1,5,3,6,2,4],[1,5,3,6,4,2],
[1,5,4,2,3,6],[1,5,4,2,6,3],[1,5,4,3,2,6],[1,5,4,3,6,2],
[1,5,4,6,2,3],[1,5,4,6,3,2],[1,5,6,2,3,4],[1,5,6,2,4,3],
[1,5,6,3,2,4],[1,5,6,3,4,2],[1,5,6,4,2,3],[1,5,6,4,3,2],
[1,6,2,3,4,5],[1,6,2,3,5,4],[1,6,2,4,3,5],[1,6,2,4,5,3],
[1,6,2,5,3,4],[1,6,2,5,4,3],[1,6,3,2,4,5],[1,6,3,2,5,4],
[1,6,3,4,2,5],[1,6,3,4,5,2],[1,6,3,5,2,4],[1,6,3,5,4,2],
[1,6,4,2,3,5],[1,6,4,2,5,3],[1,6,4,3,2,5],[1,6,4,3,5,2],
[1,6,4,5,2,3],[1,6,4,5,3,2],[1,6,5,2,3,4],[1,6,5,2,4,3],
[1,6,5,3,2,4],[1,6,5,3,4,2],[1,6,5,4,2,3],[1,6,5,4,3,2],
[2,3,1,4,5,6],[2,3,1,4,6,5],[2,3,1,5,4,6],[2,3,1,5,6,4],
[2,3,1,6,4,5],[2,3,1,6,5,4],[2,3,4,5,1,6],[2,3,4,6,1,5],
[2,3,5,4,1,6],[2,3,5,6,1,4],[2,3,6,4,1,5],[2,3,6,5,1,4],
[2,4,1,3,5,6],[2,4,1,3,6,5],[2,4,1,5,3,6],[2,4,1,5,6,3],
[2,4,1,6,3,5],[2,4,1,6,5,3],[2,4,3,5,1,6],[2,4,3,6,1,5],
[2,4,5,3,1,6],[2,4,5,6,1,3],[2,4,6,3,1,5],[2,4,6,5,1,3],
[2,5,1,3,4,6],[2,5,1,3,6,4],[2,5,1,4,3,6],[2,5,1,4,6,3],
[2,5,1,6,3,4],[2,5,1,6,4,3],[2,5,3,4,1,6],[2,5,3,6,1,4],
[2,5,4,3,1,6],[2,5,4,6,1,3],[2,5,6,3,1,4],[2,5,6,4,1,3],
[2,6,1,3,4,5],[2,6,1,3,5,4],[2,6,1,4,3,5],[2,6,1,4,5,3],
[2,6,1,5,3,4],[2,6,1,5,4,3],[2,6,3,4,1,5],[2,6,3,5,1,4],
[2,6,4,3,1,5],[2,6,4,5,1,3],[2,6,5,3,1,4],[2,6,5,4,1,3],
[3,4,1,2,5,6],[3,4,1,2,6,5],[3,4,1,5,2,6],[3,4,1,6,2,5],
[3,4,2,5,1,6],[3,4,2,6,1,5],[3,4,5,6,1,2],[3,4,6,5,1,2],
[3,5,1,2,4,6],[3,5,1,2,6,4],[3,5,1,4,2,6],[3,5,1,6,2,4],
[3,5,2,4,1,6],[3,5,2,6,1,4],[3,5,4,6,1,2],[3,5,6,4,1,2],
[3,6,1,2,4,5],[3,6,1,2,5,4],[3,6,1,4,2,5],[3,6,1,5,2,4],
[3,6,2,4,1,5],[3,6,2,5,1,4],[3,6,4,5,1,2],[3,6,5,4,1,2],
[4,5,1,2,3,6],[4,5,1,3,2,6],[4,5,1,6,2,3],[4,5,2,3,1,6],
[4,5,2,6,1,3],[4,5,3,6,1,2],[4,6,1,2,3,5],[4,6,1,3,2,5],
[4,6,1,5,2,3],[4,6,2,3,1,5],[4,6,2,5,1,3],[4,6,3,5,1,2],
[5,6,1,2,3,4],[5,6,1,3,2,4],[5,6,1,4,2,3],[5,6,2,3,1,4],
[5,6,2,4,1,3],[5,6,3,4,1,2]]
Jim FitzSimons
--- In Math4u@yahoogroups.com, "Brian Edward Jensen"
<brianejensen@...> wrote:
>
> Question 1
> I agree with Jim's answer. There are 6 color choices for one
square
> and 5 remaining color choices for the opposite square. But the
cube
> can be flipped over causing half the choices to be duplicates. So
> the answer is 6*5/2=15
> Question 2
> First of all we chose 4 colors out of 6 different colors.
> I get 6*5*4*3=360.
> Let's chose two colors to exclude:
> I get 6*5=30
> What a discrepancy! We should get the same answer picking 4
colors
> or excluding 2 colors.
> Obviously, if we chose 6 colors out of six different colors, there
is
> only one possibility. So we must chose the lower number. There
are
> 30 different ways to choose 4 colors out of 6.
> Let's call these colors 1,2,3,and 4.
> These 4 colors are to be arranged in a band around the cube. One
of
> the sides will have color 1. There are three remaining colors to
> choose from to be opposite color 1.
> So I'd say the answer is 30*3=90.
> Question 3
> We have a cube colored with 6 colors.
> There are 5 choices for the color opposite color 1. Let's call
the
> remaining colors 3,4,5,and 6. Color 3 will be between the first
two
> colors.
> There are 3 choices for the color opposite color 3.
> There are 2 remaining choices for arranging the last 2 colors.
> So I'd say the answer is 5*3*2=30
> Regards,
> Brian
>
>
> --- In Math4u@yahoogroups.com, "sanjivadayal" <sanjivadayal@>
> wrote:
> >
> > Question:-
> > Given six different colours and a cube.
> > 1. In how many ways two opposite faces of the cube can
> > be coloured with two different colours?
> > 2. In how many ways four faces of the cube can be
> > coloured with four different colours of which two
> > faces are opposite and other two faces are also
> > opposite?
> > 3. In how many ways all six faces of the cube can be
> > coloured with six different colours?
> >
>
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