Final answer:
To determine whether a permutation is even or odd, count the number of inversions it has. If the number of inversions is even, the permutation is even. If the number of inversions is odd, the permutation is odd.
Step-by-step explanation:
To determine whether a permutation is even or odd, we need to examine the number of inversions in the permutation. An inversion occurs when two elements are in the wrong order relative to each other. If a permutation has an even number of inversions, it is considered even. If it has an odd number of inversions, it is considered odd. Let's analyze each of the given permutations:
(a) (7 4 1 5 6 2 3 8) - This permutation has 16 inversions, so it is even.
(b) (71864) - This permutation has 1 inversion, so it is odd.
(c) (12)(76)(345) - This permutation is a composition of three cycles, which means it can be split into three separate permutations. The first and second cycles have no inversions, so they are both even. The third cycle has 3 inversions, so it is odd. Overall, this permutation is odd.
(d) (1276)(3241)(7812) - Each individual cycle has an odd number of inversions, so the total number of inversions is odd. Therefore, this permutation is odd.
(e) (123)(2345)(1357) - This permutation is a composition of three cycles. The first and third cycles have no inversions, so they are both even. The second cycle has 4 inversions, so it is even. Overall, this permutation is even.