Number of Ways of Threading Beads on a Loop of Wire

Theorem

Let there be $n$ beads to be threaded on a circular loop of wire.

Let $N$ be the number of different ways to thread those $n$ beads.

Then:

$N = \dfrac {\paren {n - 1}!} 2$

Proof

This is the same as Number of Ways of Seating People at Circular Table, except that having threaded the beads in one arrangement, you can get another arrangement by flipping the ring of beads over.

From Number of Ways of Seating People at Circular Table, the number of arrangements of $n$ beads on a ring, without flipping it over, is $\paren {n - 1}!$.

But every arrangement can be paired with exactly one other arrangement such that those two arrangements are effectively the same.

Hence the result.

$\blacksquare$

Sources

• 1953: L. Harwood Clarke: A Note Book in Pure Mathematics ... (previous) ... (next): $\text I$. Algebra: Permutations and Combinations: The number of ways of threading $n$ beads on a wire