SECTION 3.5 Formulas Involving Combination
 
 Simple manipulations of Combinations
 
Formula of Complementary Combinations
 
Pascal's Formula
 
Binomial Theorem
 
 


Simple manipulations of Combinations.

            We know that  there are  ways to choose r distinct elements without regard to order from a set of elements.
            In this section a number of  formulas that give values of  in special cases are discussed.

1. The value  of 

    For all integers n  >  0

   =    = = 1

Example
    Determine the value of 
=  = = 1
 
 
 
 

 2.The value of 

    For all integers  n > 0
 
             =  
                      =  
            =  
=   n 
 
 

Example
Determine the value of 
               =  
                       =  
              =  

    =      7 
 
 

3.  The value of 

     For all integers n > 0
 
    =  
 
            =  
 
    =  
 
 
 Example

 Determine the value of 
 
  =  
            =  
  =  
      =  21                 
 
 

 


 

The Formula of Complementary Combinations

  = 
For all  non negative integers n and r with r < n

   =  
 
                  =  

                                                           =  ( By interchanging factors 
                                                                                           in the denominator) 
  =  
 
 
 

 
 
 



The Pascal's Formula
    Pascal's formula is named after the seventeenth century French mathematician Blaise Pascal.
This formula is very useful  in combinatorics. This formula enables any computation of higher combinations in terms of lower ones.
 
The formula relates  the value of   to the values of  and  . By the preceeding
it  means that
 
      =  + 

 
 whenever  n and r are positive integers with  n < r . If the values of  are known , then
 values of  can be computed for all  r such that 0 < r < n

Pascal's Triangle ( values of  )
 
 
n \  r
0
1
2
3
4
5
......
r-1
r
......
 
0
1
 
 
 
 
 
 
 . 
 
 
1
1
1
 
 
 
 
 
.
 
 
2
1
2
1
 
 
 
 
.
 
 
3
1
3
3
1
 
 
 
 . 
 
 
4
1
4
6
4
1
 
 
 . 
 
 
5
1
5
10
10
5
1
 
 . 
 
 
:
:
:
:
:
:
:
:
:
:
:
:
:
:
 
:
:
:
:
.....
n
 
 
 
 
 
 
 
..... 
  
+
 
.....
n +1
 
 
 
 
 
 
 
 
 .....
 
=
 
.....
:
:
:
:
:
:
:
:
:
:
:
:
:
:
:
:
:
:
:
:
.....
 
Note:
      The value of an element (for example 10) in the Pascal's triangle is obtained by the sum of the element right above it(4) and  the element to the left of the element ( which is 6) . This is shown in the above description of the Pascal's triangle in white blocks. In the example for

 Proof of Pascal's Formula
 
 Let n and r be positive integers where r < n
 

  +  = + 
                             = +
 
 To add the above fractions the denominators are to be equalised, to achieve this
 the left fraction (both numerator and denominator)is multiplied by r and the right
 fraction (both numerataor and denominator) by  (n-r +1)
 
  +  = + 
 
                         =  + 
 
                         = 

                         = 

                         = 

                         = 
 

Example

Use the Pascal's triangle to compute the value of 
 
  =   +   =  3 + 1  = 4 
  
 
.

 



The Binomial Theorem
A sum of two terms in algebra is called a Binomial. The Binomial theorem gives an expression for the powers of a binomial,
(a+b)n  ,where n is a positive integer and a and b represent real numbers.
 
 
Binomial theorem 
Given any real numbers a and b and any non negative integer n, 
             (a + b)n   =  an-i bi 
                            =  an + an-1 b1 + a n-2 b2+..........+a 1b n-1+bn. 
 
 
 
Proof of the Binomial Theorem.(Using Combinatorics)

Let a and b be real numbers. Assume n to be an integer and its value to be atleast 1.
Then the expression (a+b)n can be expanded into products of n letters, where each
letter is either a or b. For each i=0,1,2,3.....,n, the  product

a n-ibi =  a a a ......a . b b b ..... b
              n-i factors     i factors
occurs as a term in the sum the same number of times as there are orderings of
(n-i)a's and i b's.
This number is  , which is the number of ways of choosing i positions into
which b's can be placed, the rest of the positions, that is n-i postions are filled
by a's.

The coefficient of a n-ibi  in the number of these terms,which is  .

From this we have

(a+b)n = a n-ibi
 
 Hence Proved.
 
  The Binomial theorem can also be derived algeabrically.

Example

  Find the coefficient of the given term when the expression is expanded by the binomial theorem.
        x6y3     in     (x+y)9
 
.The  coefficient of the term   x6y3  when (x+y)9  is expanded is 
  
  x6y3  = 84 x6y3 
     The coefficient is 84. 
 
 
 
 
 
 
 

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