Graph Theory/Juggling with Binomial Coefficients
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.
- 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.
where F(n) denote the nth Fibonacci number.
- Find probability of cycles of certain length in a permutation.