The Binomial Formula and Binomial Coefficients

Factorial $n$

If $n=1,2,3,...$ factorial $n$ or $n$ factorial is defined as
$n!=1\cdot2\cdot3\cdot\ldots\cdot n$
We also define zero factorial as
$0!=1$

Binomial Formula for Positive Integral $n$

If $n=1,2,3,\ldots$ then
$(x+y)^n=x^n+nx^{n-1}y+\frac{n(n - 1)}{2!}x^{n-2}y^2$ $+\frac{n(n-1)(n-2)}{3!}x^{n-3}y^3+\ldots+y^n$
This is called the binomial formula. It can be extended to other values of $n$ and then is an infinite series.

Binomial Coefficients

The result can also be written
$(x+y)^n=x^n+\binom{n}{1}x^{n-1}y+\binom{n}{2}x^{n-2}y^2+$ $\binom{n}{3}x^{n-3}y^3+\ldots+\binom{n}{n}y^n$
where the coefficients, called binomial coefficients, are given by
$\binom{n}{k}=\frac{n(n-1)(n-2)...(n-k+1)}{k!}=$ $\frac{n!}{k!(n-k)!}=\binom{n}{n-k}$

Properties of Binomial Coefficients

$\binom{n}{k}+\binom{n}{k+1}=\binom{n+1}{k+1}$
This leads to Pascal's triangle

$\binom{n}{0}+\binom{n}{1}+\binom{n}{2}+...+\binom{n}{n}=2^n$

$\binom{n}{0}-\binom{n}{1}+\binom{n}{2}-...(-1)^n\binom{n}{n}=0$

$\binom{n}{n}+\binom{n+1}{n}+\binom{n+2}{n}+...$ $+\binom{n+m}{n}=\binom{n+m+1}{n+1}$

$\binom{n}{0}+\binom{n}{2}+\binom{n}{4}+...=2^{n-1}$

$\binom{n}{1}+\binom{n}{3}+\binom{n}{5}+...=2^{n-1}$

$\binom{n}{0}^2+\binom{n}{1}^2+\binom{n}{2}^2+...+\binom{n}{n}^2=\binom{2n}{n}$

$\binom{m}{0}\binom{n}{p}+\binom{m}{1}\binom{n}{p-1}+...$ $+\binom{m}{p}\binom{n}{0}=\binom{m+n}{p}$

$(1)\binom{n}{1}+(2)\binom{n}{2}+(3)\binom{n}{3}+...$ $+(n)\binom{n}{n}=n2^{n-1}$

$(1)\binom{n}{1}-(2)\binom{n}{2}+(3)\binom{n}{3}-...$ $(-1)^{n+1}(n)\binom{n}{n}=0$

Multinomial Formula

$(x_1+x_2+...+x_p)^n=$ $\sum\frac{n!}{n_1!n_2!... n_p!}x_1^{n_1}x_2^{n_2}... x_p^{n_p}$
where the sum, denoted by $\sum$, is taken over all nonnegative integers $n_1,n_2,\ldots,n_p$ for which $n_1+n_2+\ldots+n_p=n$.