In the paper a sequence of bounded regions containing n independent identically and uniformly on Dndistributed points is considered. It is assumed that the d–dimensional volume v(Dn) is asymptotically proportional to n. Under these conditions it is shown that the number of pairs of points within a distance r>0 of each other is asymptotically normally distributed. For proving this among other things a lemma of BOLTHAUSEN is used, whereas even strong estimates for U–statistics are insufficient. The obtained result is applied for testing the hypothesis of randomness