Permutations and Combinations

Factorial:
  
  

Representation of symbols    and    :

      

  

FUNDAMENTAL PRINCIPLE OF COUNTING:

       Fundamental principle of Multiplication:   

                   If a total event can be sub-divided into two or more independent sub-events, then the number of ways in which the total event can be accomplished is given by the product of the number of ways in which each sub-event can be accomplished.

No. of ways in which the total event can be accomplished
= (No. of ways in which the 1st sub-event can be accomplished)
  × (No. of ways in which the 2nd sub-event can be accomplished)
  × (No. of ways in which the 3rd sub-event can be accomplished)
  × ....
  × ....

⇒ nE = nE1 × nE2 × nE3 × ....

Fundamental principle of Addition:  

                   If a total event can be accomplished in two or more mutually exclusive alternative events/ways, then the number of ways in which the total event can be accomplished is given by the sum of the number of ways in which each alternative-event can be accomplished.

Number of ways in which the total event can be accomplished
= (Number of ways in which the first alternative-event can be accomplished)
  + (Number of ways in which the second alternative-event can be accomplished)
  + (Number of ways in which the third alternative-event can be accomplished)
  + ....
  + ....

⇒ nE = nEa + nEb + nEc + ....

Permutation : 
                        An arrangement that can be formed by taking some or all of a finite set of things (or objects) is called a Permutation.Order of the things is very important in case of permutation.


    A permutation is said to be a Circular Permutation if the objects are arranged in the form of a circle.
^n P_r = n(n-1)(n-2)(n-3)......(n-r+1) = \frac{n!}{(n-r)!}     

Combination: 
                         A
selectionthat can be formed by taking some or all of a finite set of things( or objects) is called a Combination.denoted by ^n C_r \quad or \quad C(n,r) \quad or \quad \tbinom{n}{r}.  

Binomial theorem for a positive integral index : 


   
  ^n{C_i} are called the binomial coefficients
  {\left( {i + 1} \right)^{{\rm{th}}}} coefficient is ^n{C_i},
Note that ^n{C_i} = {\,^{n - 1}}{C_{i - 1}} + {\,^{n - 1}}{C_i}
                     ^n{C_r} = {\,^n}{C_{n - r}}. 
The general term of expansion {T_{r + 1}}, i.e.

 {T_{r + 1}} = {\,^n}{C_r}\,{x^{n - r}}{y^r}
Since we have (n + 1) terms , if n is even, there will be an odd number of terms, and thus there will be only one middle term, which would be ^n{C_{n/2}}\,{x^{n/2}}{y^{n/2}}. 

For example,

 {\left( {x + y} \right)^4} = {x^4} + 4{x^3}y + \mathop {6{x^2}{y^2}}\limits_{\scriptstyle\,\,\,\,\,{\rm{only}}\,{\rm{one}}\hfill\atop \scriptstyle{\rm{middle}}\,{\rm{term}}\hfill} + 4x{y^3} + {y^4}
If n is odd , there will be an even number of terms in the expansion, and thus there will be two middle terms, namely ^n{C_{\dfrac{{n - 1}}{2}}}\,\,{x^{\dfrac{{n + 1}}{2}}}\,{y^{\dfrac{{n - 1}}{2}}} and ^n{C_{\dfrac{{n - 1}}{2}}}\,\,{x^{\dfrac{{n - 1}}{2}}}\,{y^{\dfrac{{n + 1}}{2}}}


For example;

 {\left( {x + y} \right)^5} = {x^5} + 5{x^4}y + \mathop {10{x^3}{y^2} + 10{x^2}{y^3}}\limits_{{\rm{Two}}\,\,{\rm{middle}}\,\,\,{\rm{terms}}} + 5x{y^4} + {y^5}


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