Let ℱn be the set of random mappings ϕ : {1,…,n} → {1,…,n} (such that every mapping is equally likely). For x ε {l,…,n} the elements are called the predecessors of x. Let Nr denote the random variable which counts the number of points x ε {l,…,n} with exactly r predecessors. In this paper we identify the limiting distribution of Nr as n → ∞. If r = r(n) = o(n⅔) then the limiting distribution is Gaussian, if r ˜ Cn⅔ then it is Poisson, and in the remaining case rn−⅔ → ∞ it is degenerate. Furthermore, it is shown that Nr is a Poisson approximation if r → ∞.