We are given a set $S=\left\{x_{1}, \ldots, x_{n}\right\}$ of distinct positive integers such that $\operatorname{gcd}\left(x_{i}, x_{j}\right)= 1$ for any $i, j \in\{1, \ldots, n\}$ where $i \neq j$. What is the total number of invertible $2 \times 2$ matrices whose entries are distinct elements from the set $S$ ?
$\textbf{(Note:}$ For positive integers $a$ and $b, \operatorname{gcd}(a, b)$ denotes the greatest common divisor of $a$ and $b)$.
- $n^{4}$
- $(n-1)^{4}$
- $n^{2}(n-1)^{2} / 4$
- $n(n-1)(n-2)(n-3)$
- $n(n-1)(n-2)(n-3) / 4$ !