# Binomial Expansion

(

*x*+*y*)*=*^{n}^{n}C_{0}*x*+^{n}^{n}C_{1}*x*^{n}^{â€“1}*y*+^{n}C_{2}*x*^{n}^{â€“2}*y*^{2}+ ... +*,*^{n}C_{n}y^{n}where

*n*âˆˆ*N*=

- This expansion has (
*n*+ 1) terms. - In general, the term is given by
*T*_{r}_{+1}=â‹…^{n}C_{r}*x*^{n}^{â€“r}â‹…*y*^{r}*r*= 0, 1, 2, 3, ... ,*n.* - (
*p*+ 1)th term from end = (*n*â€“*p*+ 1)th term from beginning =*T*_{n}_{â€“p+1}.

# Properties of binomial coefficient

- Sum of two consecutive binomial coefficient,
^{n}C_{r}_{ }+^{n}C_{r}_{â€“1}=^{n}^{+1}*C*_{r} *r*=^{n}C_{r}*n*^{n}^{â€“1}*C*_{r}_{â€“1}- Ratio of two consecutive binomial coefficient,
- If
=^{n}C_{x}, then either^{n}C_{y}*x*=*y*or*x*+*y*=*n* ^{n}C_{0}+^{n}C_{1}+^{n}C_{2}+ ... += 2^{n}C_{n}^{n}^{n}C_{0}â€“^{n}C_{1}+^{n}C_{2}â€“ ...= 0^{n}C_{n}^{n}C_{0}+^{n}C_{2}+^{n}C_{4}+ ... =^{n}C_{1}+^{n}C_{3}+^{n}C_{5}+ ... = 2^{n}^{â€“1}- Putting
*x*=*i*in (1 +*x*)and comparing real and imaginary parts, we have^{n}^{n}C_{0}â€“^{n}C_{2}+^{n}C_{4}â€“^{n}C_{6}+ ... =and^{n}C_{1}â€“^{n}C_{3}+^{n}C_{5}â€“ ... = 2^{n}^{/2}sin