The aim of this paper is to give some combinatorial sums, identities and relations related to a family of combinatorial numbers and the Bernstein type basis functions. With the help of generating functions for the combinatorial numbers from this aforementioned family, we give some functional equations. By using these functional equations, we obtain several relationships including these combinatorial numbers and the Bernstein type basis functions. Moreover, we provide not only some identities, but also applications including combinatorial sums, integrals, the combinatorial numbers, the extended Bersntein basis functions, and the beta function. [ABSTRACT FROM AUTHOR]