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Binomial Theorem class 11 Notes Mathematics

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CBSE Class 11 Mathematics
Revision Notes
Chapter-8
Binomial Theorem class 11 Notes Mathematics

1. Binomial Theorem for Positive Integral Indices

2. General and Middle Terms

Binomial Theorem: The expansion of a binomial for any positive integral n is given by Binomial Theorem, which is
${{\left( a+b \right)}^{n}}{{=}^{n}}{{C}_{0}}{{a}^{n}}{{+}^{n}}{{C}_{1}}{{a}^{n-1}}b{{+}^{n}}{{C}_{2}}{{a}^{n-2}}{{b}^{2}}$ $+....+^{n}{{C}_{n-1}}a{{b}^{n-1}}{{+}^{n}}{{C}_{n}}{{b}^{n}}.$
The coefficients of the expansions are arranged in an array. This array is called Pascal’s triangle.
The general term of an expansion $~~{{\left( \text{a }+~\text{b} \right)}^{n}}\ \ is\ {{T}_{r+1}}\ \ {{=}^{n}}{{C}_{r}}{{a}^{n-r}}.{{b}^{r}}$
The general term of an expansion ${\left( {a - b} \right)^n} = {\left( { - 1} \right)^r}{.^n}{{\rm{C}}_r}.{a^{n - r}}.{b^r}$
The general term of ${\left( {1 + x} \right)^n}{ = ^n}{{\rm{C}}_r}.{x^r}$
The general term of ${\left( {1 - x} \right)^n} = {\left( { - 1} \right)^r}{.^n}{{\rm{C}}_r}.{x^r}$
In the expansion ${{\left( \text{a }+\text{ b} \right)}^{n}}$ , if n is even, then the middle term is the ${{\left( \frac{n}{2}+1 \right)}^{th}}$ term. If n is odd, then the middle terms are ${{\left( \frac{n}{2}+1 \right)}^{th}}$ and ${{\left( \frac{n+1}{2}+1 \right)}^{th}}$ terms.
${r^{th}}$ term from the end in ${\left( {a + b} \right)^n} = {\left( {n + 2 - r} \right)^{th}}$ term fromt he beginning.
Method to prove Binomial Theorem:

(a) Principle of Mathematical Induction.

(b) Combinatorial Method.

Factorial notation:
(i) $n! = 1 \times 2 \times 3 \times 4....... \times n;$ $0! = 1$
(ii) ${^n{{\rm{C}}_r} = {{n!} \over {r!\left( {n - r} \right)!}}}$
(iii) $^n{{\rm{C}}_r}{ = ^n}{{\rm{C}}_{n - r}}$
(iv) $^n{{\rm{C}}_r}{ + ^n}{{\rm{C}}_{r - 1}}{ = ^{n + 1}}{{\rm{C}}_r}$

The Binomial Theorem class 11 Notes

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