A Probabilistic Approach to Some Binomial Identities

Elementary proofs abound: simply choose x = y = 1 in the binomial expansion of (x + y). The reader is surely aware of many other proofs, including some combinatorial in nature. At the end of the previous century, the evaluation of these sums was trivialized by the work of H. Wilf and D. Zeilberger [7]. In the preface to the charming book [7], the authors begin with the phrase You’ve been up all night working on your new theory, you found the answer, and it is in the form that involves factorials, binomial coefficients, and so on, ... and then proceed to introduce the method of creative telescoping. This technique provides an automatic tool for the verification of this type of identities. Even in the presence of a powerful technique, such as the WZ-method, it is often a good pedagogical idea to present a simple identity from many different points of view. The reader will find in [1] this approach with the example