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


Source: MAA