HMMT November 2014 E7

Consider the set of 5 -tuples of positive integers at most 5 . We say the tuple ( $a_{1}, a_{2}, a_{3}, a_{4}, a_{5}$ ) is perfect if for any distinct indices $i, j, k$, the three numbers $a_{i}, a_{j}, a_{k}$ do not form an arithmetic progression (in any order). Find the number of perfect 5 -tuples.