In this paper, we give a representation of the average of complete joint weight enumerators of two linear codes of length $n$ over $\mathbb{F}_{q}$ and $\mathbb{Z}_{k}$ in terms of the compositions of $n$ and their distributions in the codes. We also obtain a generalization of the representation for the average of $g$-fold complete joint weight enumerators of codes over $\mathbb{F}_{q}$ and $\mathbb{Z}_{k}$. Finally, the average of intersection numbers of a pair of Type III (resp. Type IV) codes, and its second moment are found.