The Binomial Theorem.

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Presentation transcript:

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

(a + b)4 = (a + b)(a + b)(a + b)(a +b) =1a4 + 4a3b +6a2b2 +4ab3+1b4 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

(a + b)5 =1a5 + 5a4b +10a3b2 +10a2b3+5ab4+1b4 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 ---- 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 A symbol is introduced to represent this selection. nCr 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. nCr

nCr = 5C2 = 5C2 = 5C2 =

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

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