The set S = {1, 2, 3} consists of three elements. So 3! = 6 different permutations are possible:

{1, 2, 3}, {2, 3, 1}, {3, 1, 2}, {3, 2, 1}, {2, 1, 3}, {1, 3, 2}.

Example 2

Each of the permutations, {2, 3, 1} and {3, 1, 2}, is a sequence of two transpositions of elements of the set S = {1, 2, 3}:

{1, 2, 3} {3, 2, 1} {2, 3, 1},

{1, 2, 3} {2, 1, 3} {3, 1, 2}.

Therefore, the permutations are even.

Example 3

The permutation {3, 2, 1} is the transposition of elements 1 and 3 of the set S = {1, 2, 3}.

The number of transpositions is an odd number, and so the permutation is odd.

Example 4

The permutation {2, 3, 1} has two inversions of elements:

2 and 1, since 2 is at the left of 1 but 2 > 1, and

3 and 1, since 3 is at the left of 1 but 3 > 1.

The permutation {3, 1, 2} has two inversions of elements:

3 and 1, since 3 is at the left of 1 but 3 > 1, and

3 and 2, since 3 is at the left of 2 but 3 > 2.

Therefore, the permutations are even.

Example 5

The permutation {3, 2, 1} has three inversions of elements:

3 and 2, since 3 is at the left of 2 but 3 > 2,

3 and 1, since 3 is at the left of 1 but 3 > 1, and

2 and 1, since 2 is at the left of 1 but 2 > 1.

The permutation {2, 1, 3} has one inversion of elements:

2 and 1, since 2 is at the left of 1 but 2 > 1.

The permutation {1, 3, 2} has one inversion of elements:

3 and 2, since 3 is at the left of 2 but 3 > 2.

The permutations have odd numbers of inversions of elements, and so they are odd.