Question 12
For each permutation $a_1,$ $a_2,$ $a_3,$ $\ldots,$ $a_{10}$ of the integers $1,$ $2,$ $3,$ $\ldots,$ $10,$ form the sum $$
\begin{gathered}|a_1-a_2|+|a_3-a_4|\\+|a_5-a_6|+|a_7-a_8|\\+|a_9-a_{10}|.\end{gathered}
$$ The average value of all such sums can be written in the form $\frac{p}{q},$ where $p$ and $q$ are relatively prime positive integers. Find $p+q.$
\begin{gathered}|a_1-a_2|+|a_3-a_4|\\+|a_5-a_6|+|a_7-a_8|\\+|a_9-a_{10}|.\end{gathered}
$$ The average value of all such sums can be written in the form $\frac{p}{q},$ where $p$ and $q$ are relatively prime positive integers. Find $p+q.$