# Equality of Two Permutations

Two permutations and of degree are said to be equal if we have , .

__Example__:

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:

If then

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 coincides 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 is the set consisting of all permutations of degree . If is 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: