6.6. Combinations and permutations.
Simple counting problems allow one to list each possible way that an event can occur.
However, some events can occur in so many different ways that it would be difficult to
write out an entire list. Hence, one must use the fundamental counting principle.
Fundamental Counting Principle:
If one event can occur in m ways and a second event can occur in n ways, then
both events can occur in m· n ways, provided the outcome of the first event does
not influence the outcome of the second event.
Note: The fundamental principle of counting can be extended to any number of events.
When the outcome of one event does not influence the outcome of the other, such events
are said to be independent events.
Example: A restaurant offers a choice of 2 salads, 6 main dishes, 4 side dishes, and 3
desserts. How many different 4-course meals can be selected?
Note: Four independent events are involved: selecting a salad, selecting a main
dish, selecting a side dish, and selecting a dessert.
The first event can occur in 2 ways, the second in 6 ways, the third in 4 ways, and
the fourth in 3 ways. Thus, there are
2· 6· 4· 3 = 144 possible meals.
Example: David has 6 books that he wishes to arrange on his desk. How many different
arrangements are possible?
Note: Six events are involved: selecting a book for the first spot, selecting a book
for the second spot, and so on. The outcome of the first event does not influence
the outcome of the other events since one book has already been chosen.
David has 6 choices for the first spot, 5 choices for the second spot, 4 choices for
the third spot, and so on. Thus, there are
6· 5· 4· 3· 2· 1 = 720 different arrangements for the books.
Example: How many different 7-digit telephone numbers are possible if the first digit
can not be zero and no digit may repeat?
Note: Our number system consists of 10 digits, which are {0, 1, 2, 3, 4, 5, 6, 7, 8,
and 9}.
