We have seen in the previous chapter that (a + b)ⁿ can be written as

where we have the specific formula for the binomial coefficients:

We now look at a different interpretation of these numbers and will see why is called "n choose k." Let

Expanding this product, we get
 

The terms of this expansion correspond to the subsets of S = {x₁, x₂, x₃}. That is, we can associate the term 1 with the empty subset of S; the terms x₁, x₂, and x₃ with the singleton subsets of S; the terms x₁x₂, x₁x₃, and x₂x₃ with the doubleton subsets; and x₁x₂x₃ with S itself. (S is the only subset with 3 elements.)
 
 

Tuesday, May 12, 1998