A [i]derangement [/i] of the letters $ABCDEF$ is a permutation of these letters so that no letter ends up in the position it began such as $BDECFA$. An [i]inversion [/i] in a permutation is a pair of letters $xy$ where $x$ appears before $y$ in the original order of the letters, but $y$ appears before $x$ in the permutation. For example, the derangement $BDECFA$ has seven inversions: $AB, AC, AD, AE, AF, CD$, and $CE$. Find the total number of inversions that appear in all the derangements of $ABCDEF$.

Let's think step by step and output the final answer within \boxed{}.
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