Binomial coefficients and pascal's triangle

[Atomos]

I am working through a mathematics olympiad problem book, and I am asked to prove that n choose r, where n is the row number and r is the term number in the row is equal to that term. Can someone please give me a hint? I have not been able to find ANY proofs on the internet through a basic search, just sites that acknowledge the relationship between binomial coefficients/theorem and pascal's triangle.

 

[Tide]

It might help to recognize that each number in Pascal's triangle is equal to the total number of ways of reaching that location starting at the top and moving downward such that in going from one row to the next you can only move one step to the left or one step to the right. (Like a random walk!)

 

[Hurkyl]

Here's another way to do it:

Find some algebraic conditions that completely specify Pascal's triangle.
Prove that nCr satisfies those algebraic conditions.