It is easily seen that this pattern also holds on the line for n = 8, and that the coefficients on that line are therefore:

The binomial coefficient can be rewritten as

Similarly,
This leads us to conjecture thatholds in all cases. We prove this by mathematical induction in the following theorem.

THEOREM: If n and r are integers with  then
 

Proof: If n = 0, the only allowable value of r is 0 and= 1. Since

the formula holds for n = 0.
 
 

Monday, June 8, 1998