# 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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