Binomial Theorem

tags: Mathematics

All Mathematics Formula by Abhyas here

Please see README if this is the first time you are here.

Prerequisite

Learn Factorials from permutation and combination chapter.

Learn

${}^n{C}_r$ formula and its properties for combination

${}^n{C}_r=\frac{n!}{(n-r)!\times r!}r$

Basic

1. $(x+y{)}^n={}^n{C}_0{x}^n{y}^0{+}^n{C}_1{x}^{n-1}{y}^1{+}^n{C}_2{x}^{n-2}{y}^2{+}^n{C}_3{x}^{n-3}{y}^3+\dots {+}^n{C}_n{x}^0{y}^n$

2. $(x+y{)}^n=\sum _{r=0}^n{}^n{C}_r{x}^{n-r}{y}^r$

3. Sum of power in each term is n

4. ${}^n{C}_0$,
   ${}^n{C}_1$,
   ${}^n{C}_0$, etc are called Binomial Coefficients

5. Binomial coefficient of terms equidistintial from beginning and end are the same.

6. $n!$
   • $=n(n-1)(n-2)(n-3)\dots 3\times 2\times 1$
   • $=n\times (n-1)!$ -
     $n\in \mathbb{Z}$ i.e. only whole numbers
   • $-n!=undefined=\mathrm{\infty }$
   • thus
     $\frac{1}{-n!}=0$

7. ${}^n{C}_r=\frac{n!}{r!\times (n-r)!}=c(n,r)$ where,
   $0\le r\le n$

8. Properties of Binomial Coefficients:

   • ${}^n{C}_r\in \mathbb{Z}\mathrm{\forall }x,n\in {\mathbb{Z}}^{\mathbb{+}}$
   • ${}^n{C}_r={}^n{C}_{n-r}$ Point number 3
   • ${}^n{C}_r+{}^n{C}_{r-1}={}^{n+1}{C}_r={}^{n+1}{C}_{badaWalaR}$
     eg.
     ${}^7{C}_5{+}^7{C}_4{=}^8{C}_5$
   • ${}^n{C}_0{=}^n{C}_n=1$
   • $\frac{{}^n{C}_r}{{}^n{C}_{r-1}}=\frac{n-r+1}{r}=\frac{n-chotaR}{badaR}$
   • ${}^n{C}_r=\frac{n}{r}{}^{n-1}{C}_{r-1}=\frac{n(n-1)}{r(r-1)}{}^{n-2}{C}_{r-2}=\dots$

9. If

   ${}^n{C}_x={}^n{C}_y$ then
   $x=y$

General Term

General Term of

$(x+y{)}^n$

$=T_{r+1}={}^n{C}_r{x}^{n-r}{y}^r$

Middle Term and Highest Binomial Coefficient

Since number of terms in exansion is n+1. Thus, middle term has different cases for even and odd index.

1. Even
   $n$ in
   $(x+y{)}^n$
   Middle Term will be
   $T_{\frac{n}{2}+1}$
2. Odd
   $n$ in
   $(x+y{)}^n$
   Middle Term will be
   $T_{\frac{n+1}{2}}$ and
   $T_{\frac{n+1}{2}+1}$

Highest Binomial Coefficient
Middle Terms have the highest binomial coefficient

Number of terms in expansion

For

$(x+y+z+...uptorterms{)}^n$

Number of terms in expansion =

${}^{n+(r-1)}{C}_{r-1}$

Examples,

In

$(x+y{)}^n$, r = 2

$\therefore$ Number of terms =

${}^{n+1}{C}_1=(n+1)$ terms

In

$(x+y+z{)}^n$, r = 3

$\therefore$ Number of terms =

${}^{n+2}{C}_2$ terms

Numerically Greatest Term

Numerically Greatest between -8 and 5 is -8

Algebrically Greatest between -8 and 5 is 5

Method

In

$(x+y{)}^n$

Find

$\mid \frac{T_{r+1}}{T_r}\mid$

$\therefore \mid \frac{T_{r+1}}{T_r}\mid =\mid \frac{{}^n{C}_r{x}^{x-r}{y}^r}{{}^n{C}_{r-1}{x}^{n-r+1}{y}^{r-1}}\mid$

$\therefore \mid \frac{T_{r+1}}{T_r}\mid =\mid \frac{n-r+1}{r}\frac{y}{x}\mid$

$\therefore \mid \frac{T_{r+1}}{T_r}\mid =\frac{n-r+1}{r}\mid \frac{y}{x}\mid$

$[\because$ it is always positive

$]$

Put,

$\mid \frac{T_{r+1}}{T_r}\mid \ge 1$

Find

$r\le k$ (say)

Case I: If

$k\in \mathbb{I}$, then NGT is

$T_k$ and

$T_{k+1}$

Case II: If

$k\notin \mathbb{I}$, then NGT is

$T_{[k]+1}$

Licensing and Links

All Mathematics Formula by Abhays here

This work is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License.