2017 AIME I Problem 11

Consider arrangements of the 9 numbers 1, 2, 3, \ldots , 9 in a 3 \times 3 array. For each such arrangement, let a_1, a_2, and a_3 be the medians of the numbers in rows 1, 2, and 3 respectively, and let m be the median of \{a_1, a_2, a_3\}. Let Q be the number of arrangements for which m = 5. Find the remainder when Q is divided by 1000.

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.