The probability that <Emphasis Type="Italic">r</Emphasis> elements of a rank <Emphasis Type="Italic">n</Emphasis> free group generate a rank <Emphasis Type="Italic">r</Emphasis> subgroup

Granted the three integers n ≥ 2, r, and R, consider all ordered tuples of r elements of length at most R in the free group F _n . Calculate the number of those tuples that generate in F _n a rank r subgroup and divide it by the number of all tuples under study. As R → ∞, the limit of the ratio is known to exist and equal 1 (see 1]). We give a simple proof of this result.