On Sat, 11 Jul 2015 23:29:00 -0400, Steve Walker wrote:

Can't seem to figure out an answer. Permutation/combination stuff.
Suppose I have n red balls, x yellow balls, y green balls, and z orange
balls. (no blue. G) I need a formula for how many unique ways there
are to arrange them. Even better would be a formula for how many unique
ways to arrange them on the perimeter of a circle, so that no pattern
can be duplicated by rotating the circle. Extra credit for a link to an
algorithm to generate the patterns. Racking my brain, and Googling for
the last week.

If you had N = x + y + z unique balls*, then there would be N! (N
factorial) ways to arrange them. Within the yellow balls, there would be
x! unique ways to arrange them, but you lose that. Ditto green and
orange.

So there are

((x+y+z)!) / ((x!)(y!)(z!))

unique ways to arrange the balls in a line.

I'm in a different building from my statistics book; if I remember I'll
look it up and tell you the correct name for the function. For two
numbers x and y, the spoken expression is "x+y choose x" -- but I can't
remember the formal name to look up.

On finding the number of combinations that go away if you arrange them in
a circle -- I'm pretty sure that's hard. For a start, if x+y+z is prime
then there are no rotational symmetries to be had at all. Then, if x+y+z
and x are coprime, there are no rotational symmetries (or y, or z). But
if x+y+z and x do have common factors, the location of the yellow balls
in the circle does suddenly matter. It's enough to make me run away
screaming, or demand to be paid by the hour to figure it out.

* How unique you are if you have N 2 balls is another matter. 0 and 2
cover most of the population, 1 a small but significant fraction, but for
N 2 there is a sharply diminishing proportion -- at N 5 you're well
past "unusual" and into "astonishing" or perhaps even "frighteningly
pitiful".

--
www.wescottdesign.com