
When tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.

[ASSESS]

[ADVANCE]

[VERIFY]

[SIMPLIFY]

[SYNTHESIZE]

[PIVOT]

[OUTPUT]

You should strictly follow the format below:

[ACTION NAME]

# Your action step 1

# Your action step 2

# Your action step 3

...

Next action: [NEXT ACTION NAME]


system
Let $n\in\mathbb{N}^{*}$, for a permutation $i_{1}i_{2}\ldots i_{n}$ of $1, 2, \ldots, n$, if for $s < t$, we have $i_{s} > i_{t}$, then $(i_{s}, i_{t})$ is called an inversion of the permutation $i_{1}i_{2}\ldots i_{n}$. The total number of inversions in the permutation $i_{1}i_{2}\ldots i_{n}$ is called its inversion number. For example, for a permutation $231$ of $1, 2, 3$, there are only two inversions $(2,1)$ and $(3,1)$, so the inversion number of the permutation $231$ is $2$. Let $f_{n}(k)$ be the number of all permutations of $1, 2, \ldots, n$ with $k$ inversions.
- $(1)$ Find the values of $f_{3}(2)$ and $f_{4}(2)$;
- $(2)$ Find the expression for $f_{n}(2)$ ($n\geqslant 5$) in terms of $n$.

Present the answer in LaTex format: \boxed{Your answer}
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