## Proprties of Binomial Expansion Coefficients

The binomial coefficients, written
$\begin{pmatrix}n\\r\end{pmatrix}={}^nC_r=\frac{n!}{r!(n-r)!}$
appear if a random variable follows a binomial distribution.
If an event has a constant probability
$p$
of occurring at each attempt, then the probability of occurring
$r$
times in
$n$
attempts is
$P(r)={}^nC_r p^r(1-p)^{n-r}$
.
The coefficients
${}^nC_r$
have several important properties.
They are symmetric:
${}^nC_r={}^nC_{n-r}$
.
For example,
${}^7C_2=\frac{7!}{2!5!}=\frac{7!}{5!}{2!}={}^7C_5$
.
The sum of the binomial coefficients is a power of
$\sum^n_{r=0} {}^nC_r=2^n$
.
For example the coefficients when
$n=3$
are
${}^3C_0=1, \: {}^3C_3=3, \: {}^3C_2=3, {}^3C_3=1$
and
$1+3+3+1=8$
.
We can calculate the binomial coefficients recursively
${}^nC_r=\frac{n-r}{r+1} {}^nC_r$
.
For example
${}^7C_3=35=\frac{8-2}{2+1} \times 21={}^7C_2$
.