Using Cayley diagrams the authors establish a bijective correspondence between the set of subgroups of index \(n\) in a free group with \(k\) generators and a set of permutations in \(n+1\) symbols satisfying certain conditions. This leads to a new proof of M. Hall’s formula for the number of such subgroups [see Can. J. Math. 1, 187-190 (1949; Zbl 0031.34001)].