Question 134
Let $X$ be the set of all sequences $a = (a_1, a_2, \dots, a_{2000})$ such that $a_i \in \{0, 1, 2\}$ for $1 \le i \le 1000$ and $a_i \in \{0, 1\}$ for $1001 \le i \le 2000$. Given $a, b$ in $X$, we define the distance $d(a, b)$ between $a$ and $b$ as the number of values for $i$, $1 \le i \le 2000$, for which $a_i \ne b_i$. Find the number of distance-preserving functions $f: X \to X$, i.e., such that $d(f(a), f(b)) = d(a, b)$ for all $a$ and $b$ in $X$.