The above theorem tells us that the coefficient of is

 Since this expression has the same value when r and s are interchanged, we again see that the binomial coefficients have the symmetry relation

 
By writing out the factorials more explicitly, we see that

Cancelling common factors, we now have
 

This last formula it generally called the Binomial Theorem.
 
The formulas
 

for r > 0

enable us to extend the definition ofpreviously defined only for integers n and r with  to allow n to be any integer. We then have, for example, and 
 

Monday, June 22, 1998