Let G G be a graph consisting of m m vertex-disjoint cycles with possibly different numbers of vertices on each cycle. We want to count the number of ways of selecting k k vertices in G G such that there are exactly l l edges spanned by these k k vertices. For m = 1 m = 1 , the problem is equivalent to the Whitworth bracelet problem with two colors and a closed-form solution is known. In this paper we show that the solution for the many-cycle case can be written as a sum of the solutions for single-cycle cases.