binomial theorem

in algebra, the binomial theorem describes the algebraic expansion of exponents of a binomial

the binomial theorem says it is possible to expand the polynomial $(x+y{)}^n$ into a sum involving terms of the form $a{x}^b{y}^c$ where $b$ and $c$ are non-negative integers and $a=C(n,b)$ where $C$ is the combination function

$(x+y{)}^n=(\genfrac{}{}{0ex}{}{n}{0})x^n{y}^0+(\genfrac{}{}{0ex}{}{n}{1})x^{n-1}{y}^1+(\genfrac{}{}{0ex}{}{n}{2})x^{n-2}{y}^2+\cdots +(\genfrac{}{}{0ex}{}{n}{n-1})x^1{y}^{n-1}+(\genfrac{}{}{0ex}{}{n}{n})x^0{y}^n$

using the binomial theorem, we have the binomial expansions
the binomial coefficients are in Pascal’s triangle

$(a+b{)}^0=1$
$(a+b{)}^1=a+b$
$(a+b{)}^2=a^2+2ab+b^2$
$(a+b{)}^3=a^3+3a^{2}b+3a{b}^2+b^3$
$(a+b{)}^4=a^4+4a^{3}b+6a^2{b}^2+4a{b}^3+b^4$
$(a+b{)}^5=a^5+5a^{4}b+10a^3{b}^2+10a^2{b}^3+5a{b}^4+b^5$
$(a+b{)}^6=a^6+6a^{5}b+15a^4{b}^2+20a^3{b}^3+15a^2{b}^4+6a{b}^5+b^6$

guide to binomial expansions