Permutations and Combinations

Multiplication counting principle:

This is used to determine the number of POSSIBLE OUTCOMES when there is more than one activity/selection occurring.

MULTIPLY the # of POSSIBLE OUTCOMES for each individual event!

Examples:

1. A certain car comes in three body styles with a choice of two engines, a choice of two transmissions, and a choice of six colors. What is the minimum number of cars a dealer must stock to have one car of every possible combination?

2. Max goes through the cafeteria line and counts seven different meals and three different desserts that he can choose. Which expression can be used to determine how many different ways Max can choose a meal and a dessert?

3. A deli has five types of meat, two types of cheese, and three types of bread. How many different sandwiches, consisting of one type of meat, one type of cheese, and one type of bread, does the deli serve?

4. Cole's Ice Cream Stand serves sixteen different flavors of ice cream, three types of syrup, and seven types of sprinkles. If an ice cream sundae consists of one flavor of ice cream, one type of syrup, and one type of sprinkles, how many different ice cream sundaes can Cole serve?

Ex: The access code for a car’s security system consists of four digits.Each digit can be 0 through 9.

How many access codes are possible if:a. each digit can be repeated?b. each digit can be used only once and not repeated?c. each digit can’t be repeated and it can’t start with zero

9) How many different letter arrangements can be formed using the letters of the following words if all the letters must be used?

a) JUMP b) MISSISSIPPI c) ALABAMA

n! is read as n-factorial…it means MULTIPLY all the whole numbers less than n until you get to

1!

5! = (5)(4)(3)(2)(1) = 120

n! = n(n-1)(n-2)(n-3)…..(1)

n!

in calculator:

math > PRB > 4

PERMUTATIONS COMBINATIONS

n

! =

( )!r

nP

n r!

!( )!n r

nC

r n r

arrangement when ORDER IS IMPORTANT

arrangement when

ORDER IS NOT IMPORTANT

in calculator:

math > PRB >2

in calculator:

math > PRB >3

When should you use a Permutation   vs.     Combination?

ORDER MATTERS ORDER does NOT MATTER

Arranging numbers/lettersChoosing a teamSeating arrangments/Lining people up

Selecting a group/committee/representative

Choosing a specific position/role

1st, 2nd, 3rd place or pres. &vice-pres

Dealing cards from a deckChoosing multiple winners

1) There are 12 people on a basketball team, and the coach needs to choose 5 to put into a game. How many different possible ways can the coach choose a team of 5 if each person has an equal chance of being selected?

2) In a game, each player receives 5 cards from a deck of 52 different cards. How many different groupings of cards are possible in this game?

3) The bowling team at Lincoln High School must choose a president, vice president, and secretary. If the team has 10 members, which expression could be used to determine the number of ways the officers could be chosen?

4) A teacher wants to divide her class into groups. Which expression represents the number of different 3-person groups that can be formed from a class of 22 students?

5) Evaluate:  

6) A coach must choose five starters from a team of 12 players.  How many different ways can the coach choose the starters?

7) There are fourteen juniors and twenty-three seniors in the Service Club.  The club is to send four representatives to the State Conference.

a) How many different ways are there to select a group of four students to attend the conference?

b) If the members of the club decide to send two juniors and two seniors, how many different groupings are possible?

8) A committee of 5 members is to be randomly selected from a group of 9 teachers and 20 students. Determine how many different committees can be formed if 2 members must be teachers and 3 members must be students.