  In this paper we find exact formulas for the numbers of partitions and
compositions of an element into $m$ parts over a finite field, i.e. we find the
number of nonzero solutions of the equation $x_1+x_2+...+x_m=z$ over a finite
field when the order does not matter and when it does, respectively. We also
give an application of our results in the study of polynomials of prescribed
ranges over finite fields.
