Problem of the Day #16

POTD December 5, 2023
Author: munch

Let \pi = a_1, a_2, \cdots, a_{8} be some permutation of the numbers 1, 2, \cdots, n. We say that two elements a_i and a_j of \pi form an inversion if i < j and a_i > a_j. (For example, there are five inversions in the permutation 4312; they are 31, 32, 41, 42, and 43). Find the last 3 digits of the total number of all permutations \pi of the numbers 1, 2, \cdots, 8 such that the number of inversions in \pi is a multiple of 6.