To find the circular permutations, begin by taking one of the items as a placeholder.
Table:
Regular Permutations
Circular Permutations
2 out of 2
2
1
3 out of 3
6
2
4 out of 4
24
6
Formula: _nP_n=(n-1)!
Practice makes perfect
We are told that circular permutations are the total number of ways in which a set of objects can be arranged in a fixed circle or a loop. For example, the ways a group of friends can sit around a table. Additionally, permutations that are equivalent after a rotation are considered the same.
We can see that no matter how many places each person shifted, the permutation is the same. Therefore, we can set one person as a placeholder and arrange the other people around this person.
Table
Using the above information, we can find the number of permutations for 2 out of 2. For regular permutations, we need to calculate the factorial of 2.
2!=2*1 ⇒ 2!=2
Now, for circular permutations, we can set one of the two items as a placeholder. After this, we only have one item to place in the second position. Therefore, there is only one circular permutation when taking 2 out of 2.
Similarly, let's first find the regular permutations for 3 out of 3. To do so, let's calculate the factorial of 3.
3!=3*2*1 ⇒ 3!=6
If we set an item as a placeholder for circular permutations, we leave two available places and two items to arrange.
We can see that there are two circular permutations when taking 3 out of 3. Using the same reasoning we can find regular permutations and circular permutations when taking 4 out 4. Let's first find the factorial of 4.
4!=4*3*2*1 ⇒ 4!=24
Again, after taking one of the 4 items of as a place holder, we have three items to arrange. Note that the 3 remaining places can be filled by finding the regular permutations of 3 out of 3, which is 6. Therefore, we have 6 circular permutations. We can now create our table.
Regular Permutations
Circular Permutations
2 out of 2
2
1
3 out of 3
6
2
4 out of 4
24
6
Formula for Circular Permutations
Comparing the permutations of our table, we can see that circular permutations of 3 out of 3 are equal to regular permutations of 2 out of 2. Similarly, circular permutations of 4 out of 4 equal regular permutations of 3 out of 3. Following this reasoning, circular permutations for n out of n will equal regular permutations of n-1 out of n-1.
Circular Pemutations ofnout of n
=
Regular Permutatios ofn-1out of n-1
Now, from the regular Permutations Formula we can state that circular permutations of n out of n is given by the factorial of n-1.
Circular Permutations Formula:
_nP_n=(n-1)!
This formula is true because after taking one of the n items as a placeholder, we have to find the regular permutations of the n-1 remaining items, which is (n-1)!.
