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Binomial Coefficient: A Viewpoint

 

The formula to obtain Binomial Coefficient can be written as:

n! / r! × ( n - r )!

, which is equal to the number of combinations.

Here,

n  -> number of unique objects 

r  ->  number of objects that are                  

        considered ( r <= n  )

____________________

Suppose n = 4

Objects that are being considered:

( 1 ) A

( 2 ) B

( 3 ) C

( 4 ) D

_____________________

Let us restrict ourselves to deal with only three objects at a time. We will attempt to form combinations taking only three objects at a time.

So, here

r = 3

_____________________

From the ongoing discussion,

n = 4 and

r = 3

So, No. of combinations =

                4! / ( 3! × ( 4 - 3 )! ) = 4

Let us observe the four combinations that are formed:

___________________________

SlotI           Slot2          Slot3

___________________________

       A            B              C


       A             B              D


       B              C            D


        A              C           D

____________________________

The thought provoking  pattern of combinations tells  me to compare the objects- A, B, C, D- with tasks of Computer.  These tasks are arranged in three columns- Slot1, Slot2, and Slot3.

Tasks that occur in Slot1 have the highest rank. The Computer will firstly attend the tasks in Slot1. Tasks in Slot2 will get second priority. Slot3 tasks are to be attended at last.

_____________________________

Tasks    1st View 2nd View  3rd View

______________________________

     A         Slot I         Slot I          Slot I     


      B        Slot I         Slot II         Slot II

  

       C       Slot II        Slot II        Slot III


       D       Slot IIII      Slot III      Slot IIII   

_____________________________

Paying little attention, we can make out that the Task A always gets Slot I. Task B gets into Slot I only  for once. For the rest of the two occurences Task B is seen in Slot II. Task B and Task C appear in Slot II for  twice.  Task C is seen in  Slot III for only once. Task D always resides in Slot III.

_______________________

Observation: Combinations can serve to prioritize the participating elements. The resulting priorities can help in allocation of resources to elements in combinations. Prioritizing a list of elements forming combinations is also important as regards Computer Technology. That's it.

_______________________





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