__Finding the PON of the given
arrangement of the permutation__

**Suppose and arrangement of the
permutation is given let it be as shown below**

__ __

**This is one of the arrangement of 5P5
permutation sequence. Its PON is shown in the first Colum. **

**We have to find by formula the this
number 89 towards the corresponding arrangement of object c
e d a b**

**We know the formula of PON (object
into number)**

__Formula permuting (object into
number) __

**1+**

**1)
****(Actual Position of 1**^{st}
object-1)! x (Total number of previous objects in the **permutation arrangement**** which is right
side of the 1**^{st} object)+

**2)
****(Actual Position of 2**^{nd}
object-1)! x (Total number of previous objects in the **permutation arrangement**** which is right
side of the 2**^{nd} object)+

**3)
****(Actual Position of 3**^{rd} object
-1)! x (Total number of previous objects in the **permutation arrangement**** which is right side of
the 3**^{rd} object) +

**.**

**.**

**.**

**.**

** r) (Actual Position of last object-1)!
X (Total number of previous objects in the ****permutation arrangement**** which is right side of the
last object)**

** **

** **

** **

** **

**we know the actual positions and
linear positions of each objects**

**c
e d a b**

**3
5 4 1 2****………… (actual position of object)**

**1
2 3 4 5****…………(linear position of object)**

**We know the number of objects (r) = 5**

**Substituting the required in the formula
of PON**

__Now considering the first term of PON__

**1)
****(Actual Position of 1**^{st}
object-1)! x (Total number of previous objects in the **permutation arrangement**** which is right
side of the 1**^{st} object)+

**The actual position of 1**^{st}
object = 1

**Total number of previous objects in the ****permutation arrangement**** which is right side of 1**^{st} object = 0

**Since there is no object previous to
a **

**Substituting the value we get**

**1)
****(1-1)! x (0) = 0**

** **

__Similarly for second term of PON__

**2)
****(Actual Position of 2**^{nd}
object-1)! x (Total number of previous objects in the **permutation arrangement ****which is right
side of the 2**^{nd} object)+

** **

**The actual position of b = 2**^{nd}
object = 2

**Total number of previous objects in
the ****permutation arrangement**** which is right side of 2**^{nd} object = 0

**c
e d a b**

**As per the given arrangement of
object there is no object right side of b because b is at the end of the
arrangement. **

**Substituting the value we get.**

**2)
****(2-1)! x (0) = 0**

** **

__Similarly for the third term of PON__

**3)
****(Actual Position of 3**^{rd}
object -1)! x (Total number of previous objects in the **permutation arrangement**** which is right
side of the 3**^{rd} object) +

**The actual position of c
= 3**^{rd} object = 3

**Total number of
previous objects in the ****permutation arrangement**** which is right side of 3**^{rd} object = 2

**c****
e d a
b **

**Substituting the value we get. **

**3)
****(3 -1)! x (2) = (2)! x (2) = 2 x 2 = 4**

** **

__Similarly for the 4__^{th} term
of PON

**4)
****(Actual Position of 4**^{th}
object -1)! x (Total number of previous objects in the **permutation arrangement**** which is right
side of the 4**^{th} object) +

**The actual position of d
= 4**^{th} object = 4

**Total number of
previous objects in the ****permutation arrangement**** which is right side of 4**^{th} object =
2

**c e d a b**

**Substituting the value
we get**

**4) (4 -1)! x (2) = (3)! x
(2) = 6 x 2 = 12 **

__Similarly for the 5__^{th}
term of PON

**5)
****(Actual Position of 4**^{th}
object -1)! x (Total number of previous objects in the **permutation arrangement**** which is right
side of the 4**^{th} object)

**The actual position of
d = 5**^{th} object = 5

**Total number of
previous objects in the ****permutation arrangement**** which is right side of 5**^{th} object =
3

**c e d
a b**

**Substituting the value
we get **

**5) **** (5 -1)! x (3) = (4)! x (3) = 24 x 3 = 72**

**We have found numbers
corresponding to each and every term now substituting as per the formula we
get.**

**The formula of PON**

**1+**

**1)
****(Actual Position of 1**^{st}
object-1)! x (Total number of previous objects in the **permutation arrangement**** which is right
side of the 1**^{st} object)+

**2)
****(Actual Position of 2**^{nd}
object-1)! x (Total number of previous objects in the **permutation arrangement**** which is right
side of the 2**^{nd} object)+

**3)
****(Actual Position of 3**^{rd}
object -1)! x (Total number of previous objects in the **permutation arrangement**** which is right
side of the 3**^{rd} object) +

**.**

**.**

**.**

**.**

** r) (Actual Position of last object-1)!
X (Total number of previous objects in the ****permutation arrangement**** which is right side of the
last object)**

**= 1 + 0 + 0 + 4 + 12 + 72 = **__89__

**This is the proof of the PON formula. **