Abstract: The dependence on the sum of bivariate random vectors with Farlie-Gumbel-Morgenstern copulas is studied in the paper. Firstly, the Kendall's and the Spearman's on two independent random vectors' sum with the copulas are deduced, and the specific equation with exponential marginal distribution is shown. Then, the proposition is proved that there exists no tail-dependence under some conditions on marginal distribution. Finally, we calculate some numerical instances for different marginal distributions by using Monte Carlo method. The conclusions and methods in this paper have theoretical significance for the dependence between two random indices of the combination of enterprise, and lay foundations for the further study.