Proprties of Binomial Expansion Coefficients

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