Pascal’s rule proof


We need to show

(nk)+(nk-1) = (n+1k)

Let us begin by writing the left-hand side as

n!k!(n-k)!+n!(k-1)!(n-(k-1))!

Getting a common denominator and simplifying, we have

n!k!(n-k)!+n!(k-1)!(n-k+1)! = (n-k+1)n!(n-k+1)k!(n-k)!+kn!k(k-1)!(n-k+1)!
= (n-k+1)n!+kn!k!(n-k+1)!
= (n+1)n!k!((n+1)-k)!
= (n+1)!k!((n+1)-k)!
= (n+1k)
Title Pascal’s rule proof
Canonical name PascalsRuleProof
Date of creation 2013-03-22 11:47:14
Last modified on 2013-03-22 11:47:14
Owner akrowne (2)
Last modified by akrowne (2)
Numerical id 10
Author akrowne (2)
Entry type Proof
Classification msc 05A10
Classification msc 81T13
Classification msc 53C80
Classification msc 82-00
Classification msc 83-00
Classification msc 81-00