So the other day I had a quiz on permutations and combinations. Easy stuff right? We have all heard the formula for the number of permutations of $n$ objects around a circle; $(n-1)!$ of course! Let us first take a look at why this is so.
The number of ways to arrange $n$ objects in a line is clearly $n!$. The only thing that is different in a circle is that since a circle is cyclic, the starting point does not make any difference. Thus the $n!$ arrangements in a line are split into equivalence classes depending on the starting point, which are of course disjoint. Each arrangement has $n$ equivalent arrangements, depending on the starting point. Thus we have $n! / n = (n - 1)!$ different arrangements for objects arranged in a circle.
And permutations where objects are repeated? No biggie, it’s
\[\frac{n!}{n_1!\cdot n_2!\cdots n_k!},\]where $n_1, n_2, \dots$ are the number of objects of the first kind, second kind, etc, because we have $n!$ arrangements in the line, out of which $n_1$ objects of the first kind can be permuted in $n_1!$ ways which would still be the same arrangement, $n_2$ objects of the second kind can be permuted in $n_2!$ ways, and so on.
What if we combined the 2 problems, permutations around a circle with repeated objects? Still doesn’t seem like a difficult proposition, does it? One would logically think it to be
\[\frac{(n - 1)!}{n_1!\cdot n_2!\cdots n_k!},\]especially if we look at the logic behind both the formulae: $(n - 1)!$ ways to arrange around a circle, out of which $n_i$ similar objects can be permuted in $n_i!$ ways; seems intuitively correct.
Turns out … it’s not. Or more accurately, not always.
About the author
Swastik Patnaik is a student of IISER Mohali. This particular article placed second in our Article Writing Contest, 2022.