In this paper, as a generalization of the binomial random graph model, we define the model of multigraphs as follows： let G（n; {pk}） be the probability space of all the labelled loopless multigraphs with vertex set V = {v1, v2, ..., vn }, in which the distribution of tvi,vj, the number of the edges between any two vertices vi and vj is P{tvi,vj =k}=Pk, k=0, 1,2,...and they are independent of each other. Denote by Xd = Xd（G）,Yd = Yd（G）, Zd = Zd（G） and Zcd = Zcd（G） the number of vertices of G with degree d, at least d, at most d and between c and d. In this paper, we discuss the distribution of Xd, Yd, Zd and Zcd in the probability space G（n; （Pk））.
