Average of an expression over all permutations

Let $Q_{j}$ represent the $j'th$ arrangement of all values of $a_{i}$ .
There are $200!$ permutations of these values. Notice that the average value of $Q$ can be thought of as the sum of $Q_{i}$ over all permutations divided by the total number of permutations
ie. $avg(Q) =\frac{\displaystyle\sum_{j=1}^{200!} Q_{j}}{200!}$ .
let $s_{1}$ be the sum over all permutations of $\mid a_{1} - a_{2} \mid$ , $s_{2}$ be sum over all permutations of $\mid a_{3} - a_{4} \mid$ , etc.
Then, $\displaystyle\sum_{j=1}^{200!} Q_{j} = s_{1} +s_{2}+s_{3}+...+s_{99}+s_{100}$
however, notice that $s_{1}=s_{2}=s_{3}=...=s_{99}=s_{100}$ because each case that shows up in one term for some $j$ in $Q_{j}$ also shows up as a case in another term for some other $j$ .
Thus, $\displaystyle\sum_{j=1}^{200!} Q_{j} =100 s_{1}$ , and $avg(Q) = \frac{100 s_{1}}{200!}$
To find $s_{1}$ , consider summing all of the cases of $\mid u - v \mid$ where $u > v$ , each case will happen in $2 \times 198!$ of the 200! permutations, since there are $198!$ permutations of the other 198 values of $a_{i}$ and $\mid u-v \mid =\mid v-u \mid$ ... Thus $s_{1} = 2 \times 198! \times D$ where D is the value of this sum
if you write out terms in $D$ as an array of values, and group terms into rows based on equivalent values, it becomes easy to notice the value 199 comes up once, 198 comes up twice, 197 three times, etc.
Thus $D = 1 \times (199) +2 \times (198) + 3 \times (197) +...+198 \times (2) + 199 \times (1)$
In sigma notation $D= \displaystyle\sum_{k=1}^{199} k (200-k) =\displaystyle\sum_{k=1}^{199} (200k - k^{2}) =200 \displaystyle\sum_{k=1}^{199} k -\displaystyle\sum_{k=1}^{199} k^{2}$
Now using the fact that $\displaystyle\sum_{k=1}^{n} k = \frac{n(n+1) }{2}$ and $\displaystyle\sum_{k=1}^{n} k^{2} = \frac{n(2n+1)(n+1)}{6}$
$D = 200 \times \frac{199(200)}{2} - \frac{199(200)(399)}{6} = \frac{200(199)}{2} (200- \frac{399}{3}) =\frac{200(199)}{2} (67)$
plugging this value into $s_{1} = 2 (198!) D$ gives $s_{1} = 200! ( 67 )$
recall $avg(Q) = \frac{100 s_{1} }{200!}$ subbing in our numerical value for $s_{1}$ gives us $avg(Q) = \frac{100 \times (200!) \times 67}{200!} = 100 \times 67 =$ 6700