Problem 1 [24 pts]: Permutations and inversions For each of these problems, write down the pemutation/combination eapression. You do not have to simplify. For example P(5, 2) + () is a valid answer. Consider sequence A = (1,2,3,4, 5, 6, 7). i. How many permutations of A do not have have either 1 or 2 at the beginning, at the end, or in the exact-middle ? E: . How many permutations of A contain the subsequence (1,2,3) in this original order (not necessarily in consecutive positions) ? H. H- -C . For a permutation of A, an inversion is a pair (a,b) that is not in the original order: 3. comes before b in the permutation, but b was before a in the original order. (If they get moved around but are still in the correct order relative to each other, they are not inverted.)] For example, if we permute A into (1,2,3,4,5,7,6) has one inversion: (7,6). What pairs are inversions in the permutation B = (6,4, 1, 7,5,2, 3)? How many inversions? iv. What is the maximum number of inversions a permutation of A can have ? Which permutation achieves that