Till now, we have learnt permutations as arrangement of objects in different orders in a straight line. Now, we are going to learn permutations when objects are arranged in different orders in a circle, or a chain.
Suppose there are five objects a, b, c, d and e. In linear permutations, they can be arranged in 5! = 120 different ways. When we arrange them in a circle, as follows:
The following arrangements are all considered the same: (because there is no starting or ending point)
Because these are circular permutations, and we have to keep in mind that in a circle, you start counting from any one point, the order of the objects will not be different, since there is no starting point and no ending point.
Circular permutations are the permutations of objects when they are arranged in a circular fashion in which there is no starting point and ending point.The main difference between circular and linear permutations is that in circular permutations there is no starting or ending point. This results in the loss of permutations since from wherever you start counting in a circle, the order of the objects will be the same, since there is no fixed starting point and ending point. Let us understand this with the help of an example:
Suppose there are five objects a, b, c, d and e. In linear permutations, they can be arranged in 5! = 120 different ways. When we arrange them in a circle, as follows:
The following arrangements are all considered the same: (because there is no starting or ending point)
- a, b, c, d, e
- b, c, d, e, a
- c, d, e, a, b
- d, e, a, b, c
- e, a, b, c, d
Because these are circular permutations, and we have to keep in mind that in a circle, you start counting from any one point, the order of the objects will not be different, since there is no starting point and no ending point.
We know that the number of permutations of 5 different objects taken all at a time is 5! = 120 when they are arranged in a straight line.
But when the 5 objects will be arranged in a circle, some of the permutations that were considered before will not be considered as valid permutations now. This is because in a circle we can start counting from anywhere and the different arrangements obtained by counting from different objects in any particular arrangement will not be really "different", as they would represent the same circular arrangement of objects overall.
For 5 objects arranged as a, b, c, d and e in a circle, the following all arrangements are the same:
- a, b, c, d, e
- b, c, d, e, a
- c, d, e, a, b
- d, e, a, b, c
- e, a, b, c, d
Thus, for every arrangement of the 5 objects in a particular order, there will be 5 such arrangements that will be the same in a circular permutations view.
Therefore, the number of circular permutations of 5 different objects will be 5!/5.
Now we can derive the formula for circular permutations:
If there are 'n' number of different objects, and they are arranged in a circular fashion, then the total number of circular permutations of these 'n' objects is equal to the number of linear permutations of the 'n' objectsdivided by 'n'.
Circular permutations of 'n' objects = n!/n
= (n(n - 1)! )/n
= (n - 1)!
Again, this is because for every other linear permutation that is there, there will exist 'n' such linear permutations that will be considered the same in the light of circular permutations.
No comments:
Post a Comment