Let (Formula presented.) and (Formula presented.) be two monomial ideals, where A and B are two polynomial rings with disjoint variables. Considering a general set-up of monomial filtrations, we study the behavior of the (Formula presented.) -number under binomial expansion. As an application, we get an explicit formula of (Formula presented.) in terms of (Formula presented.) and (Formula presented.), where (Formula presented.) denotes the k-th symbolic power of an ideal L. Furthermore, an analogous formula is extended for the (Formula presented.) -number of integral closure of (Formula presented.).