The N-order coupled nonlinear Schr dinger equation with variable coefficients is studied and its 3-soliton solutions are obtained. The soliton interaction is studied by the asymptotic analysis and graph analysis. The results show that when the eigenvalues are different, the 3-soliton solutions can be expressed as the regular solitons, the bounded solitons, and the combination of them, respectively. Under the certain conditions, both the regular bright solitons and the bounded bright solitons can realize the elastic and the inelastic interactions, respectively. However, the dark solitons only have the inelastic interaction. But for the combination of the regular and the bounded solitons, the interacting rules of the bright soliton components are very complex and significantly influenced by the value of the parameters, but the dark soliton components still maintain the elastic interaction.