1996 AIME Problem 12

For each permutation a_1,a_2,a_3,\cdots,a_{10} of the integers 1,2,3,\cdots,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 \dfrac{p}{q}, where p and q are relatively prime positive integers. Find p+q.

Leading zeroes must be inputted, so if your answer is 34, then input 034

Full credit goes to MAA for authoring these problems. These problems were taken on the AOPS website.