What proof is there for the formula for Pascals triangle?
Pascal’s triangle is simply a pictorial but working model of a basic mathematical concept which has applications is several different areas eg permutations and combinations, polynomials from the expansion of (p + q)^n, the binomial probability distribution etc. For these a formula which generates the basic coefficients a(n, k) of the Pascal triangle is not only required but essential. It may be written as a(n, k) = n!/[(n – k)! k!] where n, k are positive integers and n! is the factorial 1 x 2 x 3 x …. (n – 1) x n The problem with providing a proof is really that of knowing where to start. For example, some readers will have recognised the above expression as equivalent to the number of ways k items can be selected from a population of n distinct objects. If you are happy with that, Pascal’s Triangle follows fairly simply. If not, I need to provide an explanation of the source of the above expression. Alternatively, the various equations can be proved by induction; it’s a formal math
