EXAMPLE 38 Traveling to some but not all of the cities

Example 37 calculated the number of possible routes for traveling to cities. However, suppose we are interested in traveling to some but not all of the cities? For example, suppose that the salesman is traveling to three of the eight cities. Find the number of possible routes.

Solution

Eight choices are available for the first city, seven choices for the second city, and six choices for the third city. The salesman is traveling to three cities only, so the number of possible routes is thus

This result may be rewritten using factorial notation, as follows: