Counting PrinciplesCounting Principles Counting PrinciplesCounting Principles

Chapter 3.4Chapter 3.4

Objectives• Use the Fundamental Counting Principle

to find the number of ways 2 or more events can occur

• Find the number of ways a group of objects can be arranged in order

• Find the number of ways to choose several objects from a group without regard to order

• Use counting principles to find probabilities

The Fundamental Counting Principle

• If 1 event can occur in m ways and a second event can occur in n ways, the number of ways the 2 events can occur in sequence is m*n.

• This rule can be extended for any number of events occurring in sequence.

Buying a car• You can choose a Ford, Subaru, or

Porsche• You can choose a small or medium

car• You can choose red (R), purple (P),

white(W) or green (G)• How many choices do you have?• 3*2*3 = 18 choices

You can draw a tree diagram to show this

Security Systems• The access code for a car’s security

system consists of 4 digits. Each digit can be 0 through 9.

• How many access codes are possible if each digit can be used only once and not repeated?

• 10*9*8*7• = 5040

Security Systems• The access code for a car’s security

system consists of 4 digits. Each digit can be 0 through 9.

• How many access codes are possible if each digit can be repeated?

• 10*10*10*10• = 10,000

License Plates• How many license plates can you

make if a license plate consists of• Six (out of 26) letters each of

which can be repeated?• Six (out of 26) letters each of

which can not be repeated?

Permutations• A permutation is an ordered

arrangement of objects. • The number of different

permutations of n distinct objects is n!

• n! is read “n factorial”• 5! = 5*4*3*2*1• 0! is defined as 1

Baseball example• How many starting lineups are

possible for a 9 player baseball team?

• 9*8*7*6*5*4*3*2*1 • = 362,880• Where is the factorial key on • your calculator?

Finding nPr

• Find the number of ways of forming 3-digit codes in which no digit is repeated:

• Select the first digit – (10 choices)

• Select the second digit – (9 choices for each of the possible 10 first choices =

90)

• Select the third – (8 choices for each of the 90 previous choices = 720)

Permutations of n Objects taken r at a

Time• The number of permutations of n

distinct objects taken r at a time is

• nPr = n! , where r ≤ n

• (n-r)!• Read this “Probability of n choose

r”

Permutations of n Objects taken r at a

Time• Use the formula for the last

problem

• nPr = n! , where r ≤ n

• (n-r)!

• 10P3 = 10! =10*9*8*7*6*5*4*3*2*1

• (10-3)! 7*6*5*4*3*2*1

Permutations of n Objects taken r at a

Time

Another example . . .

AAAB• How many ways can this be

arranged?• AAAB• AABA• ABAA• BAAA• 4

AAAABBC• How many ways can this be

arranged?• 105, but it would be tedious

writing these all out, so let’s use a formula:

Distinguishable Permutations

Try it with AAAB:• 4! = 4*3*2*1 = 4• 3!*1! 3*2*1

Try it with AAAABBC• 7! = 7*6*5 = 105• 4!*2!*1! 2

Combinations• You want to buy 3 hats from a selection

of 5. How many possible choices do you have?

• ABC ABD ABE• ACD ACE• ADE• BCD BCE

• BDE CDE

10 Choices

Combinations• A combination is a selection of r objects

from a group of n objects without regard to order and is denoted by nCr.

• The number of combinations of r objects selected from a group of n objects is

• nCr = = n! .

• (n-r)!r!

Combinations• Try this with the 5 hats, choose 3:

• nCr = = n! .

• (n-r)!r!

• nCr = 5! . = 5*4*3*2*1 = 10

• (5-3)!*3! 2*1*3*2*1

MISSISSIPPI• A word consists of 1 M, 4 I’s, 4 S’s, and 2 P’s.

If the letters are randomly arranged in order, what is the probability that the arrangement spells out Mississippi?

• How many distinguishable permutations?• 11! =• 1!*4!*3!*2!• 34,650• Probability(Mississippi) = 1/34,650 = .000029

Diamond Flush• Find the probability of being dealt 5 diamonds

from a standard deck of cards.• How many ways to choose 5 out of 13 diamonds?

• 13C5

• How many ways to choose a 5 card hand?

• 52C5

• P(diamond flush) = 13C5 = 1,287 = .0005

• 52C5 = 2,598,960

Homework• P. 157 12-40 every 4th problem

Now try this . . .• With Skittles!