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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