Next we replace each of x₁, x₂, and x₃ by x in our two expressions for M. This results in
 

Thus we see thatfor k = 0, 1, 2, 3 is the number of ways of choosing a subset of k elements from a set S of 3 elements. Similarly, one can see that the number of ways of choosing k elements from a set of n elements is 
For example, the set {1, 2, 3, 4, 5} with 5 elements has subsets having 3 elements. Since

it is not too difficult to write out all ten of these subsets as
 

If we drop the braces enclosing the elements of each subset, the resulting sequence is said to be a combination of 3 things chosen from the set {1, 2, 3, 4, 5}. Thus the ten combinations of 3 things chosen from this set of 5 objects are
 
 

Note that changing the order in which the objects of a combination are written does not change the combination. For example, 1, 2, 4 is the same combination as 1, 4, 2.
 

Tuesday, May 12, 1998