How many polynomials in $Z_{p}[x]$ have degree n or less?

Question

For your reference, $Z_{p}[x]$ refers to the set of all polynomials with coefficients integer mod p. To me it seems like this and the degree (power) of the two polynomials are unrelated. What theorem would you use to figure this out?

Answer 1

The answer is very simple: a polynomial $P$ of degree less than $n$ can be identified to the set of its coefficients $(a_{0},a_{1},...a_{n})$ (with eventually $a_{n}=0$ if $P$ has degree less than $n$).

You have $p$ possibilities for each $a_{i}$ and you must make this choice $n+1$ times, so the answer is $p^{n+1}$