A detailed description of the binomial theorem and an application to permutation binomials over finite fields

In this paper, we present a detailed description of the binomial theorem and obtain some new classes of combinatorial identities. As an application, we discuss a class of permutation binomials over finite fields $${\mathbb {F}}_q$$ , which is of the form $$x^{\mu +\nu }+2x^{\mu }$$ , where $$q\equiv 1\pmod {3}$$ and $$(\nu , q-1)=\frac{q-1}{3}$$ .