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You have 6 basket balls - 3 brown, 2 red, and 1 green. What is the probobility of blindly putting them in semetrical order....

you cant - 0% or 0/100

if you have an odd # of basket balls, then one HAS to go in the middle - if there is also another odd #, then you need two different spots for middle - nope! sorry!

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You seem to be speaking of "symmetrical along just one pre-defined axis", a pre-drawn line of reflection.

Don't think 1-dimensional, think 2d. Think of a line of tape dividing a gym in half. If you lined them up in a line perpindicular to the reflection line, then yes you cannot have symmetry, but you never said they have to be lined up. In fact if you're "placing them blindly", the odds of "lining them up" are miniscule at best, unless you mean putting them in a line without looking at color. Aaanyway:

you can put all the balls in a vertical line straight on top of the tape. Or you could do 1 ball of each of the colors that have an odd # of balls along the line of symmetry, then balance the even ones around it on other side. There are many other ways of course

If you are restricted to a linear progression of balls with the line of symmetry slicing it into two halves, then you cannot have symmetry because you have to split odd-excess into halves, but you cannot do that with multiple odd-excess balls unless they are stacked on top of each other (if the line of symmetry is height-wise) or next to each other along the depth axis if the line of symmetry is depth-wise

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There are several kinds of symmetry - translation, reflection and rotation are three.

And rotational has three axes and an infinite number of angles: 180, 120, 90, 72, 60, 45, 30, 15, 12 ... etc degrees.

UR suggests that certain symmetries are attainable - e.g. stack them vertically and you can rotate about that axis any number of degrees while leaving the configuration unchanged.

As UR points out, the OP seems to be talking about reflection or 180 degree rotational symmetry about an axis perpendicular to a 1-dimensional ordering of the six balls. Restricted to one dimension, reflection or 180-degree rotation symmetry can't be achieved.

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