We are given the polynomial ring $\mathbb{Z}/3\mathbb{Z}$. We are asked to determine how many polynomials of degree $n$ there are. First of all, the possible coefficients are $0, 1,2$, if we have a polynomial of the form: $$ a_0 + a_1 X + a_2 X^2 +a_3 X^3 + \dots a_n X^n$$ There are $n+1$ degrees of freedom, namely $a_0, a_1, a_2 \dots$ up to and including $a_n$. Since there are $3$ possible values, there are $3^{n+1}$ polynomials. However, to be a degree $n$ polynomial, we should not allow $a_n=0$ this restriction would give $ 2 \cdot 3^n$.
Is this a valid deduction? I need it in the next question to actually list all of them and determine which ones are irreducible.
Briefly, yes, your argument sounds good!