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
\[{}^7C_3=35=\frac{8-2}{2+1} \times 21={}^7C_2\]
.