EDIT: This is nothing new. I found a remark to the same effect on Wikipedia.

Recently I was trying to visualize Pascal's triangle as a kind of a graph, that can be traversed from the root down. It struck me that the coefficient in a particular node/vertex not only expresses the sum of the coefficients of its immediate neighbors from the row above (the elementary school construction of the triangle), but it's also the number of unique trajectories leading from the root to that vertex. This, in turn, led me to think of a recursive definition of \(\binom{n}{k}\) that calculates the value by "walking" these trajectories.

As a reminder, the binomial coefficient is defined as follows: \[ \binom{n}{k} = \frac{n!}{(n-k)!k!} \] In CS/statistical settings it is also frequently called choose, because it describes the number of ways to choose an unordered set of \(k\) items out of \(n\). The way the values grow for increasing \(n\) can be visualized using Pascal's triangle (the \(k\) th item in the \(n\) th row is given by \(\binom{n}{k}\)):

0:                  1         
1:                1   1        
2:              1   2   1       
3:            1   3   3   1      
4:          1   4   6   4   1     
5:        1   5  10   10  5   1    
6:      1   6  15   20  15  6   1   
7:    1   7  21   35  35  21  7   1   
8:  1   8  28   56  70  56  28  8   1

Just as the elementary school relation defines a coefficient in terms of its neighbors one row up, one can think that in order to know the number of trajectories leading to a vertex one needs to add the number of trajectories leading to its left and right neighbor one row above. Using this knowledge we can define the following function (Elisp/Common Lisp):

(defun choose (n k)
  (if (or (zerop k) (= k n) (< n 2))	; Handle edges of the triangle
      1
    (+ (choose (1- n) (1- k))
       (choose (1- n) k))))

And, sure enough, one can verify that it does in fact produce correct values e.g. by reconstructing part of Pascal's triangle with it:

ELISP> (loop
        for n from 0 to 6
        do
        (princ (loop
                for k from 0 to n
                collect (choose n k)))
        (terpri))
(1)
(1 1)
(1 2 1)
(1 3 3 1)
(1 4 6 4 1)
(1 5 10 10 5 1)
(1 6 15 20 15 6 1)

nil