The Binomial Theorem The Binomial Theorem Binomial expansion
The Binomial Theorem
The Binomial Theorem (Binomial expansion) (a + b)1 = 1 a +1 b (a + b)2 = (a + b) coefficient =1 a 2 + 2 ab + 1 b 2 (a + b)3 = (a + b)(a + b) =1 a 3 + 3 a 2 b +3 ab 2 +1 b 3
The Binomial Theorem (Binomial expansion) (a + b)4 = (a + b)(a +b) =1 a 4 + 4 a 3 b +6 a 2 b 2 +4 ab 3+1 b 4 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 =1 a 5 + 5 a 4 b +10 a 3 b 2 +10 a 2 b 3+5 ab 4+1 b 4
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 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. C r
n. C r = 5 C 2 = C 5 2=
Theorem of Combination C = C n r n n-r e. g. 10 C 6 = 10 C 4
The Binomial Theorem (Binomial expansion) (a + b)5 =1 a 5 + 5 a 4 b +10 a 3 b 2 +10 a 2 b 3+5 ab 4+1 b 4 (a + b)5 =1 a 5 + 5 C 1 a 4 b +5 C 2 a 3 b 2 +5 C 3 a 2 b 3+5 C 4 ab 4+5 C 5 b 4 (a + b)n =1 an + n. C 1 an-1 b +n. C 2 an-2 b 2 +n. C 3 an-3 b 3+…. +n. Cran-rbr+…. +1 bn where n is a positive integer
- Slides: 14