Section: 11.6 / Counting Principles, Permutations and Combinations / Semester 2 - Day 47
The Fundamental Counting Principle
Do you sometimes feel yourself complaining that you never have enough clothes to wear?
Suppose you have 3 pairs of jeans, 4 shirts and 2 pairs of shoes. How many different outfits do you truly have?
Options in Ordering Lunch
(This problem was taken from a previous SAT Exam)
A cafeteria has a lunch special consisting of soup or salad, a sandwich, coffee or tea, and a dessert. If the
menu lists three soups, 2 salads, 8 sandwiches and 7 desserts, how many different lunches can you choose?
(Note: Two lunches are different if they differ in any aspect.)
Telephone Numbers in the United States
Telephone numbers in the United States begin with three-digit area codes followed by seven-digit local telephone numbers. Area codes and local telephone numbers cannot begin with 0 or 1. How many different telephone numbers are possible?
Permutations
ACTIVITY
Select 3 students to stand before the class. How many different ways can they line up facing the class?
A permutation occurs when
- No item is used more than once
- The order of arrangment in important
Using the Formula for Permutations
A corporation has seven members on its board of directors. In how many ways can it select a president, vice-president, secretary and treasurer?
Using the Formula for Permutations
You need to arrange seven of your favorite books on a small shelf. How many different ways can you arrange the books, assuming that the order of the books makes a difference to you?
Combinations
ACTIVITY
Select 3 students to stand before the class. How many different ways can a 2-person study group be assembled?
A combination occurs when
- Items are selected from the same group
- No item is used more than once
- The order of the items makes no difference
Distinguishing Between Permutations and Combinations
For each of the following problems, determine whether the problem is one involving permutations or combinations. (It is not necessary to solve the problem.)
a. Six students are running for student government president, vice-president and treasurer. The student with the greatest number of votes becomes the president, the second highest vote-getter becomes vice-president, and the student who gets the third largest number of votes will be treasurer. How many different outcomes are possible foe these positions?
b. Six people are on a board of supervisors for your neighborhood park. A three-person committee is needed to study the possibility of expanding the park. How many different committees can be formed from the six people?
c. Baskin-Robbins offers 31 different flavors of ice cream. One of their items is a bowl consisting of three scoops of ice cream, each a different flavor. How many such bowls are possible?
d. How many ways can you select 6 free DVDs from a list of 2000 DVDs?
e. In a race in which there are 50 runners and no ties, in how many ways can the first three finishers come in?
We can develop of formula for by computing permutations and combinations. Consider the letters
A, B, C and D. The number of permutations of these four letters taken three at a time is
Here are the 24 permutations:
Because the order of the items makes no difference in determining a combination, each column of six permutations represents a single combination. Thus, there is a total of four combinations:
Notice that there are 6 (or 3!) times as many permutations as combinations for this example.
This idea will hold true for all problems. There are r! more permutations that combinations.
Using the Formula for Combinations
In poker, a person is dealt 5 cards from a standard 52-card deck. The order in which you are dealt the 5 cards does not matter. How many different 5-card poker hands are possible?