Two permutations and of degree are said to be equal if we have , .
If and are two permutations of degree 4, then we have. Here we see that both and replace 1 by 2, 2 by 3, 3 by 4, and 4 by 1.
If is a permutation of degree , we can write it in several ways. The interchange of columns will not change the permutation. Thus, we can write
Therefore, if and are two permutations of the same elements of degree , then it is always possible to write in such a way that the first row of coincide with the second row of .
Total Number of Distinct Permutations of Degree :
If is a finite set having distinct elements, then we shall have distinct arrangements of the elements of . Therefore there will be distinct permutations of degree . If be the set consisting of all permutations of degree . If be the set containing of all permutations of degree then the set will have distinct elements.
This set is called the symmetric set of permutations of degree . Sometimes it is also denoted by .
Thus, ( is a permutation of degree ).
The set of all permutation of degree 3 will have 3!, i.e., 6 elements. Obviously