5.4 Counting Methods

Objectives:By the end of this section, I will beable to…

1) Apply the Multiplication Rule for Counting to solve certain counting problems.

2) Use permutations and combinations to solve certain counting problems.

3) Compute probabilities using combinations.

SAT problems

In how many ways can you pick….

Counting Rule

An easy way to find total possibilities or the SAMPLE SPACE

Counting Rule

MULTIPLY

LICENSE PLATES

In Delaware, license plates have seven characters. How many possibilities are there total? What if you could only use numbers? What if you could only use letters? What if you can’t repeat letters? What if you do not have any restrictions at

all?

10·10·10·10·10·10·10=

10,000,00026·26·26·26·26·26·26=

8,031,810,176

26·25·24·23·22·21·20=

3,315,312,000

36·36·36·36·36·36·36=

7.836 E 10

PASSWORDS

Most internet sites force you to have a password.

If one site requires 8 characters, letters only, how many possibilities are there?

26·26·26·26·26·26·26·26=

2.088 E 11

CELL PHONES

How many phone numbers are possible in Delaware?

Are there enough possibilities if everyone had their own cell phone? (There are about 850,000 people living in Delaware)

1·1·1·8·10·10·10·10·10·10=

8,000,000

(302)…-….

PERMUTATIONSAny arrangement of distinct objects in a particular order.

Factorial Notation

1) 3! = 3 x 2 x 1

2) 10! = 10 x 9 x 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1

ON YOUR CALCULATOR

MATH – PRB – !

Check this out!

PERMUTATIONS THE MOVIE

PERMUTATION FORMULA

!

( )!n r

nP

n r

n = total

r = how many you are selecting

Rankings

In how many different ways can you arrange the following beautiful ladies?

SportsIn how many ways can

teams finish in the NFC East?

Starting Line UPIn how many ways can Coach

Dinardo choose his starting offense from a total of 32 players?

CombinationAn arrangement of objects

where order is NOT important.

Combination FormulaThe number of combinations of

n total things taken r at a time is denoted as

!

!( )!n r

nC

r n r

Angel always dresses well. He is deciding which 2 shirts to wear to school this week out of his 13 clean shirts.

In how many ways can Angel choose which two shirts to wear?

Serpe is cleaning his room and actually putting his collection of baseball cards away. He usually puts 4 new cards on display each month.

In how many ways can he choose the 4 cards from a possible 32 cards?

Permutation vs. Combination

Both questions ask “in how many ways…”

Permutations need ORDERRanking

Combinations DON’TMore random

1) The senior class is going to take the top four students (by GPA) to Jamaica for a little R&R. In how many ways can they be chosen?

2) What if they pick the four students randomly from a hat. In how many ways can they be chosen?

Poker!How many different poker hands

consisting of five cards can be dealt from a deck of 52 cards?

What is the probability of being dealt a royal flush in five card poker?

PASCAL’s TRIANGLE It is a mathematical relationship between

numbers that can be used for computing the number of possible combinations of n things taken r at a time.

Use Pascal’s Triangle to find the following.

5 52 3

76 6 0

. .

. .

a C b C

c C d C

1 6 15 1520 6 1