Pascal’s rule (bit string proof)

This proof is based on an alternate, but equivalentMathworldPlanetmathPlanetmathPlanetmathPlanetmath, definition of the binomial coefficientMathworldPlanetmath: (nr) is the number of bit strings (finite sequencesPlanetmathPlanetmath of 0s and 1s) of length n with exactly r ones.

We want to show that


To do so, we will show that both sides of the equation are counting the same set of bit strings.

The left-hand side counts the set of strings of n bits with r 1s. Suppose we take one of these strings and remove the first bit b. There are two cases: either b=1, or b=0.

If b=1, then the new string is n-1 bits with r-1 ones; there are (n-1r-1) bit strings of this nature.

If b=0, then the new string is n-1 bits with r ones, and there are (n-1r) strings of this nature.

Therefore every string counted on the left is covered by one, but not both, of these two cases. If we add the two cases, we find that

Title Pascal’s rule (bit string proof)
Canonical name PascalsRulebitStringProof
Date of creation 2013-03-22 12:23:03
Last modified on 2013-03-22 12:23:03
Owner vampyr (22)
Last modified by vampyr (22)
Numerical id 5
Author vampyr (22)
Entry type Proof
Classification msc 05A10