Bionomial Theoram

Bionomial Theoram

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Bionomial Theoram

Introduction

Binomial Theoram is one chapter in which you see a question and then realise that it can easily be done in 2 or 3 ways. Definitely one of the most interesting chapters which uses concepts from calculus, trignometry, complex number prognessiong and wkhat not Binomial theoram finds use in number theory as well. So, let us prober deeper into this section.

Binomial theorm
Definition of Bionomial theoram
If n is a +ve integer and x₁ y are 2 complex number then .
(x + y)ⁿ ⁿ C_r x^n-r y^r

= ⁿC_oXⁿ - ⁿC₁ X^n-1 y + ⁿC₂ X^n-2 y² +............+ ⁿC_n yⁿ

Similarly (x - y) ⁿ = ⁿCoXⁿ - ⁿC₁ X^n-1 y + ⁿC₂ X^n-2 y² +.......+ (-1) ⁿ ⁿC_n yⁿ

The oefficients ⁿ C_o, ⁿC₁, .................ⁿC_n are called Bionomial coeff.
are called Bionomial coeff.

Some important facts
(i) There are (n + 1) tevms in the expression .

(ii) lthe sum of in dices. Of x and y in each term of expansion is n .

(iii) The bionomial 3efficent of the terms equidistant from begining and the end are equal .
(iv) The r-th term from end in (x + y)
ⁿ
= (h + r + 2) th term from beginning .

General term
.

In the expansion of (x + y)ⁿ the (r + 1^tn) term from beginning of expansion is
Tr + 1 = ⁿC_r a^n-r b^b

Illustration 1

Expand by biobomial theoram .

Ans T₇ from end of (a + b)ⁿ is same as T₇ from beginning of (b + a)ⁿ

of (a + b)ⁿ

or

or

or ( 2 ^1/3 3 ^1/3 ) ^{n - 12}=

=>

or

So, n = 9