Let S denote the set of all 4 element permutations of 123456

Let S denote the set of all 4 element permutations of {1,2,3,4,5,6}. Define an equivalence relation R on S as follows: two permutations x and y of s are related if they both permute the same elements. Determine the size of the equivalence class S₂₃₅₆ containing the permutation 2356.

Solution

Since the order of the number is not important, we will have

6 * 5 * 4 * 3 = 360 classes

The size of the equivalence class will be S2356

will be equal to 4 * 3 * 2 * 1 = 24

Since the order is not important

The possibilities are

{2,3,5,6}, {2,3,6,5}, {2,5,3,6},{2,5,6,3}, {2,6,3,5}, {2,6,5,3} -> starting with 2

{3,2,5,6}, {3,2,6,5}, {3,5,2,6},{3,5,6,2},{3,6,2,5},{3,6,5,2} -> starting with 3

{5,2,3,6},{5,2,6,3},{5,3,2,6},{5,3,6,2},{5,6,2,3},{5,6,3,2} -> starting with 5

{6,2,3,5}, {6,2,5,3}, {6,3,2,5},{6,3,5,2},{6,5,2,3},{6,5,3,2}->starting with 6

Hence there will be 24 elements in the equivalence class