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The Binomial Theorem

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					The Binomial Theorem
        The Binomial Theorem
         (Binomial expansion)
(a + b)1 = 1a +1b
                                coefficient
(a +   b)2 =   (a + b)(a + b)
          =1a2 + 2ab + 1b2
(a + b)3 = (a + b)(a + b)(a + b)
          =1a3 + 3a2b +3ab2 +1b3
 The Binomial Theorem (Binomial expansion)
(a + b)4 = (a + b)(a + b)(a + b)(a +b)
        =1a4 + 4a3b +6a2b2 +4ab3+1b4
 Take out the coefficients of each expansion.




                        1
 The Binomial Theorem (Binomial expansion)
  Can you guess the expansion of (a + b)5
     without timing out the factors ?


                 +      +      +     +



(a + b)5 =1a5 + 5a4b +10a3b2 +10a2b3+5ab4+1b4
The Binomial Theorem (Binomial expansion)
Points to be noticed :
• Coefficients are arranged in a Pascal triangle.
• Summation of the indices of each term is equal to
  the power (order) of the expansion.
• The first term of the expansion is arranged in
  descending order after the expansion.
• The second term of the expansion is arranged in
  ascending order order after the expansion.
• Number of terms in the expansion is equal to the
  power of the expansion plus one.
The Notation of Factorial and Combination

Factorial
---- the product of the first n positive
integers
      i.e. n! = n(n-1)(n-2)(n-3)….3×2×1
           0!is defined to be 1.
      i.e. 0!= 1
                     Combination
There are 5 top students in this class. If I
would like to select 2 students out of these
five to represent this class. How many ways
are there for my choice?
List of the combinations ( order is not considered) :
(1,2), (1,3), (1,4), (1,5), (2,3), (2,4), (2,5), (3,4), (3,5), (4,5)
A symbol is introduced to represent this
selection.

                    n Cr or nCr or Cnr
             n!
nCr =
         r!(n  r )!
             5!
5 C2 =
         2!(5  2)!
            5!
5 C2 =
          2!(3)!
           5  4  3  2 1
5 C2 =                         10
         (2  1)  (3  2 1)
 Theorem of Combination

       nCr = nCn-r
 e.g. 10C 6 = 10C4

nCr-1 + nCr = n+1Cr
Exercise 4.2
  P.108
   The Binomial Theorem (Binomial expansion)
(a + b)5 =1a5 + 5a4b +10a3b2 +10a2b3+5ab4+1b4

(a + b)5 =1a5 + 5C1a4b +5C2a3b2 +5C3a2b3+5C4ab4+5C5b4


(a + b)n =1an + nC1an-1b +nC2an-2b2
             +nC3an-3b3+….+nCran-rbr+….+1bn

                where n is a positive integer
The Binomial Theorem (Binomial expansion)


    general term in the expansion =

              Cr   a n-rbr
             n

            (r + 1)th term
 Theorem of Combination

       nCr = nCn-r
 e.g. 10C 6 = 10C4

nCr-1 + nCr = n+1Cr
Exercise 4.3
  P.114
The Binomial Theorem (Binomial expansion)

Extension to Trinomial
(1 – x + x2)4
= [1-x(1-x)]4
= 14 -4C1(1)3x(1-x) + 4C2(1)2x2(1-x)2
 -4C3(1)1x3(1-x)3 + x4(1-x)4
Exercise 4.4
  P.119

				
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