Counting Rules
The fundamental counting principle states that when making successive choices the total number of choices is the product
(number of first choices)(number of second choices left after the first choice has been made)(number of choices left after the first two choices have been made) and so on.
Example 1: In how many different orders can five different objects be put in a line?
You have five choices for the first object
then four choices left for the second object
then three choices for the third object
then two choices for the fourth object
and finally one choice left for the fifth object
The total is 5·4·3·2·1 = 120
Note: The product 5·4·3·2·1 is a special product called a factorial. It is written 5! and pronounced "five factorial".
Note: By definition 0! = 1
Example 2: Calculate ten factorial.
10! = 10·9·8·7·6·5·4·3·2·1 = 3,628,800
An arrangement of n objects is a of the objects. There are n! permutations of n objects.
Example 3: In a race with seven runners in how many ways can you award gold, silver and bronze?
By the fundamental counting principle you can award gold in 7 ways, then silver in 6 ways and bronze in 5 ways for a total of 7·6·5 = 210 ways
We call the number of different ways of awarding the medals the number of permutations of 7 objects 3 at a time. Its symbol is 7P3.
To calculate nPr on the TI83+ enter n, press MATH, select PRB from the top menu, then select nPr, then enter r and press enter.
To calculate 8P3 you should see 8 nPr 3 on the display. The answer is 336.
Combinations
Example 4: Consider the problem of selecting a committee of 3 people from a pool of seven people. If you applied the fundamental counting principle you would select the same committee in multiple ways.
For example, if Archibald, Bertha and Constance were chosen then ABC, ACB, BAC, BCA, CAB, CBA would all give the same committee. In fact each committee would be selected in 3! ways
To get the number of different committees you divide 7P3 by 6. When you make selections where to order does not matter you have a combination.
We call the number of different ways selecting the committee the number of combinations of 7 objects 3 at a time. Its symbol is 7C3.
To calculate nCr on the TI83+ enter n, press MATH, select PRB from the top menu, then select nCr, then enter r and press enter.
To calculate 8C3 you should see 8 nCr 3 on the display. The answer is 56.
Note: In problems you will often be faced with choosing between nPr and nCr . If different orders lead to different selections use P. If different orders count as the same selection then use C
Example 5. In how many ways can you choose a jury of 7 men and 5 women from a pool of 13 men and 12 women?
It is a combination because no matter in which order the jury is chosen it is the same jury.
You can choose the men in 13C7 ways and the women in 12C5 ways. Then by the fundamental counting principle the total number of ways is 13C7·12C5 =1,359,072 ways..