Answer by Ross Millikan for The number of (at most) $n-1$ distinct integers $1\le x_1,\dots,x_k\le p$ is at most $n^kp^{n-1}$

$p^{n-1}$ is a reasonable approximation which is accurate as long as $n \ll p$. The correct value is ${p \choose n-1}=p(p-1)(p-2)\ldots (p-n+2)$. Having chosen the acceptable values for the $x_j$s, each of them has $n-1$ possibilities, so you would have $(n-1)^k$ ways to assign them, which they are probably approximating by $n^k$.