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- Statistics - Discussion
Statistics - Odd and Even Permutation
Consider X as a finite set of at least two elements then permutations of X can be divided into two category of equal size: even permutation and odd permutation.
Odd Permutation
Odd permutation is a set of permutations obtained from odd number of two element swaps in a set. It is denoted by a permutation sumbol of -1. For a set of n numbers where n > 2, there are ${\frac {n!}{2}}$ permutations possible. For example, for n = 1, 2, 3, 4, 5, ..., the odd permutations possible are 0, 1, 3, 12, 60 and so on...
Example
Compute the odd permutation for the following set: {1,2,3,4}.
Solution:
Here n = 4, thus total no. of odd permutation possible are ${\frac {4!}{2} = \frac {24}{2} = 12}$. Following are the steps to generate odd permutations.
Step 1:
Swap two numbers one time. Following are the permutations obtainable:
Step 2:
Swap two numbers three times. Following are the permutations obtainable:
Even Permutation
Even permutation is a set of permutations obtained from even number of two element swaps in a set. It is denoted by a permutation sumbol of +1. For a set of n numbers where n > 2, there are ${\frac {n!}{2}}$ permutations possible. For example, for n = 1, 2, 3, 4, 5, ..., the even permutations possible are 0, 1, 3, 12, 60 and so on...
Example
Compute the even permutation for the following set: {1,2,3,4}.
Solution:
Here n = 4, thus total no. of even permutation possible are ${\frac {4!}{2} = \frac {24}{2} = 12}$. Following are the steps to generate even permutations.
Step 1:
Swap two numbers zero time. Following is the permutation obtainable:
Step 2:
Swap two numbers two times. Following are the permutations obtainable: