It's Time For Algebraic Probability!

$f_4\left(x_1, x_2, x_3, x_4\right) = \sum\limits_{1 \leq i < j \leq 4} \left(\left\lceil \dfrac{x_i}{x_j}\right\rceil + \left\lfloor \dfrac{x_i}{x_j}\right\rfloor + \left\lceil \dfrac{x_j}{x_i}\right\rceil + \left\lfloor \dfrac{x_j}{x_i}\right\rfloor\right)$

Consider the function above, where the domain for each of $$x_1, x_2, x_3, x_4$$ is $$\{1,2,3,4\}$$.

What is the probability that out of all possible choices in the domain, $$\left( x_1, x_2, x_3, x_4\right)$$ makes $$f_4$$ the minimum?


Bonus: Generalize this for $$f_n$$, where $$n \geq 2$$ and the domain is $$\{1, 2, \dots, n\}.$$

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