Filby

There have been several questions recently about arranging objects in a circle. Typically these are about arranging n red chairs and n blue chairs around a circular table. Most recently the question was about 8 objects (four of one kind and four of another) and after several varied answers and methods the conclusion was that the answer was 10 but there was no generalised method of solving this problem.

Obviously the number of ways of arranging 2n objects in a line is (2n)!

As there is a group of n all the same we can divide this by n! and again by another n! because of the other identical set of n objects

From this I would propose that the number of ways of arranging n red chairs and n blue chairs in a straight line is (2n)!/(n! x n!) = (2n)!/(n!)^2

All that is left to do is to eliminate the number of duplicates when the straight lines are turned into circular arrangements. I have no idea why but the answer seems to be given by dividing by (2n – 1)

(2n)!/((n!)^2(2n-1))

This is correct for

n = 2, 3, 4, 5 (number of arrangements 2, 4, 10, 28)

For n = 6 the formula predicts 84 although the answer given here to that question was 80

Can anyone shed any light on whether this formula is sound?

When you pour boiling water from a kettle it sparkles quite differently from cold water. If you pour it on to a hard surface it makes a harder sound. Even very hot water is not the same - only boiling. Any ideas?