Algebra 2: Section 12.1Algebra 2: Section 12.1

The Fundamental Counting Principle and Permutations

The Fundamental Counting Principle and Permutations

Examples for types of problemsExamples for types of problems

Experimenttoss a coinroll a dietoss two coinsdraw a cardselect a day of the week

Experimenttoss a coinroll a dietoss two coinsdraw a cardselect a day of the week

Sample Spaceo head or tailo 1, 2, 3, 4, 5, or 6o HH, HT, TH, or TTo 52 possible outcomeso Sun, Mon, Tues, Wed, Thurs, Fri, or Sat

Sample Spaceo head or tailo 1, 2, 3, 4, 5, or 6o HH, HT, TH, or TTo 52 possible outcomeso Sun, Mon, Tues, Wed, Thurs, Fri, or Sat

ExampleExample

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?

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?

12 7 3 252

Fundamental Counting PrincipleFundamental 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

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

The Fundamental Counting RuleThe 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

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

ExampleExample

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?

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?

5 5 5 5 5 3125

5 4 3 2 1 120

Factorial DefinitionFactorial Definition

n! = n•(n-1)•(n-2)•…•3•2•1

Examples: 4!

10!

1! = ____ 0! = ____

n! = n•(n-1)•(n-2)•…•3•2•1

Examples: 4!

10!

1! = ____ 0! = ____

= 4•3•2•1 = 24

= 3,628,800= 10•9•8•7•6•5•4•3•2•1

11

ExampleExample

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?

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?

:Letters

:Numbers261010 10 10 26 26 26 17,576,000

10 9 8 26 25 24 11,232,000

PermutationsPermutations

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?

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?

!

( )!n r

nP

n r

5 3P5!

(5 3)!

5!

2!

5 4 3 2 1

2 1

5 4 3 60

Permutation ExamplePermutation 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?

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

8 3P

8 7 6 336

8!

(8 3)!

8!

5! 8 7 6 5 4 3 2 1

5 4 3 2 1

The Permutation Rule ExampleThe Permutation Rule Example

Ex: In how many ways can a batting order be made from a team of 17 softball players?

Ex: In how many ways can a batting order be made from a team of 17 softball players?

17 9

17!

(17 9)!P

17 16 15 14 13 12 1110 9 8 7 6 5 4 3 2 1

8 7 6 5 4 3 2 1

8,821,612,800

17 16 15 14 13 12 1110 9

The Fundamental Counting RuleThe 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

Extra Example: How many possible ways can three cards

be drawn from a deck, without replacing them?

52•51•50 = 132,600

Permutations ExamplesPermutations 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

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

Fundamental Counting PrincipleFundamental 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

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

Factorial DefinitionFactorial Definition

n! = n•(n-1)•(n-2)•…•3•2•1

Examples: 4!

10!

n! = n•(n-1)•(n-2)•…•3•2•1

Examples: 4!

10!

= 4•3•2•1 = 24

= 3,628,800= 10•9•8•7•6•5•4•3•2•1

!Can use your calculator

PermutationsPermutations

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?

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?

!

( )!n r

nP

n r

5 3P5!

(5 3)!

5!

2!

5 4 3 2 1

2 1

5 4 3 60

!Can use your calculator

ExampleExample

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

MATH

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

MATH4 LETTERS

0 !None are repeated

4 4P 24

Permutations with RepetitionPermutations with RepetitionThe number of distinguishable

permutations of n objects where one object is repeated q1 times, another is repeated q2 times, and so on is:

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

1 2

!

!• !•...• !k

n

q q q

Permutations with Repetition ExamplePermutations with Repetition Example

How many distinguishable permutations of the letters in BASKETBALL are there?

How many distinguishable permutations of the letters in BASKETBALL are there?

10 LETTERSB AL

222

10!

2! 2! 2!

453,600

Permutations with Repetition ExamplePermutations with Repetition Example

How many distinguishable permutations of the letters in MISSISSIPPI are there?

How many distinguishable permutations of the letters in MISSISSIPPI are there?

11 LETTERSI S P

442

11!

4! 4! 2!

34,650