Graph Theory/Juggling with Binomial Coefficients

Introduction edit

Fluency with binomial coefficients is a great help in combinatorial arguments about graphs. You should learn to juggle with binomial coefficients as easily as you juggle with normal algebraic equations.

Techniques edit

  • Substitute specific values in the general equations, e.g. careful choice of x and y in:
 
  • Sledgehammer proof using recursion formula. You may find it helpful to 'chase' binomial coefficients on a diagram of Pascal's triangle.
  • Differentiation of a previous identity to get a new one.
  • Combinatorial arguments about permutations and combinations.

Worked Examples edit

Example: 2n

To get:

 

Put   in

 

The Identities edit

 
 
 
 
 
 


 
 
 
 
 

where F(n) denote the nth Fibonacci number.

 
 
 
 
 .
 

Applications edit

  • Find probability of cycles of certain length in a permutation.