The number of polynomials of a given degree over a finite field with value sets of a given cardinality

Let Vf denote the value set (image) of a polynomial f Fq[x]. We relate the number of polynomials f Fq[x] of degree q−1 such that Vf=k to the solutions (x1,…,xk) of a linear equation over Fq, with the added restriction that xi≠xj whenever i≠j. Using this we find a simple formula for the number of such polynomials.