What are we doing today?

• Need a calculator • Notes: Basic Combinatorics• Go over quiz• Homework

Definitions

• Independent Events:– the outcome of one event does not affect the

outcome of any other event.

• Dependent Events:– the outcome of one event does affect the

outcome of another event.

Basic Counting Principle• Suppose one event can be chosen in p

different ways and another independent event can be chosen in q different ways. Then the two events can be chosen successively in pq ways.

• This can be extended to any number of events, just multiply the number of choices for each event.

ExampleHow many sundaes are possible if you can only choose one from each of the following categories?

ice cream flavors: chocolate, vanilla, strawberry, rocky road

sauce: hot fudge, caramel

toppings: cherries, whipped cream, sprinkles

(4)(2)(3) = 24 different sundaes

Example

How many different license plates can be made if each plate consists of 2 digits followed by 3 letters followed by 1 digit?

Unless told otherwise, always assume all letters of the alphabet, all digits 0-9, and repetition is allowed. Treat each space as an event.

(10)(10)(26)(26)(26)(10)=17,576,000 possible combinations.

ExampleA test consists of 8 multiple choice questions. How many ways can the 8 questions be answered if each question has 4 possible answers?

(4)(4)(4)(4)(4)(4)(4)(4)

= 48

= 65,536

Factorials n!

definition: product of consecutive numbers from 1 to n.

Example 8! = (8)(7)(6)(5)(4)(3)(2)(1)

= 40,320

PermutationsAn arrangement of objects in a specific order or

selecting all of the objects.

The number of permutations of n objects taken r at a time, denoted P(n,r) or nPr, is

!

( )!

n

n r

ExampleThere are ten drivers in a race. How many outcomes of first, second, and third place are possible?

10 3

10!

(10 3)!P

= 720 ways

ExampleThere are 30 students in the Art Club, how many ways can the club select the President, Vice President, and Secretary for the club?

30P3 = 24,360 ways

Combinations• An arrangement of objects in which order does not

matter.

• Difference between permutations and combinations:– Combinations: grouping of objects– Permutation: putting objects in specific places or positions,

or selecting all of the objects.

!( , )

!( )!n r

nC n r or C

r n r

Permutations vs Combinations• Select a committee of 5 people from a group of 33

people.– Combination (order doesn’t matter)

• Elect a President, Vice President, Treasurer, & Secretary from a group of 40 people.– Permutation (putting in specific places)

• Pick your favorite soda, and your second favorite soda from a group of 8 sodas.– Permutation (putting in specific places)

• Buy 3 types of soda at Giant from a group of 30 sodas.– Combination (order doesn’t matter)

• Arrange the entire set of 12 books on a shelf.– Permutation (arranging all the objects)

Example

In a study hall of 20 students, the teacher can send only 6 to the library. How many ways can the teacher send 6 students?

20 6C 38,760 ways

Example

• Jessie is at the library and wants to sign out 8 books but she can only sign out 3. How many ways can she choose which books to sign out?

8 3C = 56 ways

Counting Subsets of an n-setA local pizza shop offers patrons any combination of up to 10 different toppings. How many different pizzas can be ordered if patrons can choose any number of toppings (0 through 10)?

Each topping can be seen as a yes or no question so each has 2 options:

210 = 1024 different pizzas are possible.(This includes no toppings and picking all toppings)

Homework

Pg 708: 1-8, 11-22

SAT Math II take end of this year