Binomial Theorem - Revision Notes

 CBSE Class 11 Mathematics

Revision Notes
Chapter-8
BINOMIAL THEOREM

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   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
  (a+b)n=nC0an+nC1an−1b+nC2an−2b2${\left(a+b\right)}^n{=}^n{C}_0{a}^n{+}^n{C}_1{a}^{n-1}b{+}^n{C}_2{a}^{n-2}{b}^2$+....+nCn−1abn−1+nCnbn.$+....{+}^n{C}_{n-1}a{b}^{n-1}{+}^n{C}_n{b}^n.$
• The coefficients of the expansions are arranged in an array. This array is called Pascal’s triangle.
• The general term of an expansion   (a + b)n  is Tr+1  =nCran−r.br${\left(\text{a }+\text{b}\right)}^{n}is{T}_{r+1}{=}^n{C}_r{a}^{n-r}.b^r$
• The general term of an expansion (a−b)n=(−1)r.nCr.an−r.br${\left(a-b\right)}^n={\left(-1\right)}^r{.}^n{\mathrm{C}}_r.a^{n-r}.b^r$
• The general term of (1+x)n=nCr.xr${\left(1+x\right)}^n{=}^n{\mathrm{C}}_r.x^r$
• The general term of (1−x)n=(−1)r.nCr.xr${\left(1-x\right)}^n={\left(-1\right)}^r{.}^n{\mathrm{C}}_r.x^r$
• In the expansion (a + b)n${\left(\text{a }+\text{ b}\right)}^n$, if n is even, then the middle term is the  (n2+1)th${\left(\frac{n}{2}+1\right)}^{th}$term. If n is odd, then the middle terms are (n2+1)th${\left(\frac{n}{2}+1\right)}^{th}$and (n+12+1)th${\left(\frac{n+1}{2}+1\right)}^{th}$ terms.
• rth$r^{th}$ term from the end in (a+b)n=(n+2−r)th${\left(a+b\right)}^n={\left(n+2-r\right)}^{th}$ term fromt he beginning.
• Method to prove Binomial Theorem:

         (a) Principle of Mathematical Induction.

         (b) Combinatorial Method.

• Factorial notation:
  (i) n!=1×2×3×4.......×n;$n!=1\times 2\times 3\times \mathrm{4.......}\times n;$     0!=1$0!=1$
  (ii) nCr=n!r!(n−r)!${}^n{\mathrm{C}}_r=\frac{n!}{r!\left(n-r\right)!}$
  (iii) nCr=nCn−r${}^n{\mathrm{C}}_r{=}^n{\mathrm{C}}_{n-r}$
  (iv) nCr+nCr−1=n+1Cr