There is a relationship between the number of cities one might go to and the number of possible routes to get to those cities. For example, the number of possible routes between three cities is two routes; for four cities, there are six routes; for five cities, there are twenty-four routes. You can observe a  pattern here. The table below shows that the number of routes forms a pattern in relationship to the number of  of cities. On the next page, your will learn how to select the shortest route while planning for your trips. For additional information of pattern in math, click here to go to the 'Pascal's Triangle' page.

 Number of Cities Number of Possible Routes Pattern A Pattern B 3 2 2! 2 ´ 1 4 6 3! 3 ´ 2 ´ 1 5 24 4! 4 ´ 3 ´ 2 ´ 1 6 120 5! 5 ´ 4 ´ 3 ´ 2 ´ 1 7 720 6! 6 ´ 5 ´ 4 ´ 3 ´ 2 ´ 1 8 5,040 7! 7 ´ 6 ´ 5 ´ 4 ´ 3 ´ 2 ´ 1

If Tom wants to go from Los Angeles to San Francisco, San Diego, and Las Vegas, and back to Los Angeles again, how many possible routes can he take?

Look at chart above. Four cities = six paths. He has six possible routes.

David is going on a concert tour from his hometown of San Francisco to Boston, New York, Los Angeles, Las Vegas, San Diego, Houston, London, Tokyo, Hong Kong, and then Boise. He must return home at the end of each concert to refill his suitcase with his things that he needs. How many different routes can he travel to these cities?