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