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Combi with arrangements

Source: Moldova TST 2020

March 7, 2020
combinatorics

Problem Statement

Let nn, (n≥3)(n \geq 3) be a positive integer and the set AA={1,2,...,n1,2,...,n}. All the elements of AA are randomly arranged in a sequence (a1,a2,...,an)(a_1,a_2,...,a_n). The pair (ai,aj)(a_i,a_j) forms an inversioninversion if 1≤i≤j≤n1 \leq i \leq j \leq n and ai>aja_i > a_j. In how many different ways all the elements of the set AA can be arranged in a sequence that contains exactly 33 inversions?
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