1. The Fundamental Counting Principle and Permutations
Algebra 2: Section 12.1
The Fundamental Counting Principle and Permutations

2. Examples for types of problems
Experiment
toss a coin
roll a die
toss two coins
draw a card
select a day of the week
Sample Space
head or tail
1, 2, 3, 4, 5, or 6
HH, HT, TH, or TT
52 possible outcomes
Sun, Mon, Tues, Wed, Thurs, Fri, or Sat

3. Example
You have 12 shirts, 7 pairs of pants, and 3 pairs of shoes. How many different combinations of outfits can you possibly have?
You have 12 shirts, 7 pairs of pants, and 3 pairs of shoes. How many different combinations of outfits can you possibly have?

4. Fundamental Counting Principle
If one event occurs in m ways and another occurs in n ways, then the number of ways that both events can occur is mn.
This also extends to more than two events
Ex: mnp for three events that occur in m, n, and p ways

5. The Fundamental Counting Rule
How many possible ZIP codes are there?
10•10•10•10•10 = 100,000
How many possible ZIP codes are there if 0 can not be used for the first digit?
9•10•10•10•10 = 90,000

6. Example
If there are five students who can receive five different scholarships, how many different ways can the five scholarships be awarded?
How many different ways can the scholarships be awarded if each student can only win one scholarship?

7. Factorial Definition n! = n•(n-1)•(n-2)•…•3•2•1 Examples: 4! 10!
1! = ____
0! = ____
= 4•3•2•1
= 24
= 10•9•8•7•6•5•4•3•2•1
= 3,628,800

8. Example
The standard configuration for a New York license plate is 3 digits followed by 3 letters.
How many different license plates are possible if digits and letters can be repeated?
How many different license plates are possible if digits and letters cannot be repeated?

9. Permutations An ordering of objects
Placing a certain number of objects in a certain number of positions
Notation: nPr
(n things placed in r positions)
Example:
How many ways can you place 5 different students in 3 different chairs?

10. Permutation Example
There are 8 different people running in the finals of the 400M dash. How many different ways can they place 1st, 2nd, and 3rd?

11. The Permutation Rule Example
Ex: In how many ways can a batting order be made from a team of 17 softball players?

12. The Fundamental Counting Rule
Extra Example:
How many possible ways can three cards be drawn from a deck, without replacing them?
52•51•50 = 132,600

13. Permutations Examples
Extra Example:
Six runners are in the first heat of the 100 meter sprint. In how many different ways can the race end?
6•5•4•3•2•1 = 720 ways
In how many ways can the first three spots of the race be filled?
6•5•4 = 120 ways

14. Fundamental Counting Principle
If one event occurs in m ways and another occurs in n ways, then the number of ways that both events can occur is mn.
This also extends to more than two events
Ex: mnp for three events that occur in m, n, and p ways

15. Factorial Definition n! = n•(n-1)•(n-2)•…•3•2•1 Examples: 4! 10!
= 4•3•2•1
= 24
= 10•9•8•7•6•5•4•3•2•1
= 3,628,800

16. Permutations An ordering of objects
Placing a certain number of objects in a certain number of positions
Notation: nPr
(n things placed in r positions)
Example:
How many ways can you place 5 different students in 3 different chairs?

17. Example
How many distinguishable permutations of the letters in MATH are there?
MATH

18. Permutations with Repetition
The number of distinguishable permutations of n objects where one object is repeated q1 times, another is repeated q2 times, and so on is:

19. Permutations with Repetition Example
How many distinguishable permutations of the letters in BASKETBALL are there?

20. Permutations with Repetition Example
How many distinguishable permutations of the letters in MISSISSIPPI are there?