5.5Counting Techniques

More Challenging StuffThe classical method, when all

outcomes are equally likely, involves counting the number of ways something can occur

This section includes techniques for counting the number of results in a series of choices, under several different scenarios

Example● If there are 3 different colors of

paint (red, blue, green) that can be used to paint 2 different types of toy cars (race car, police car), then how many different toys can there be?

● 3 colors … 2 cars … 3 • 2 = 6 different toys

● This can be shown in a table or in a tree diagram

Table A table of the different possibilities

This is a rectangle with 2 rows and 3 columns … 2 • 3 = 6 entries

Tree DiagramA tree diagram of the different

possibilities

Red

Blue

Green

Blue Race Car

Blue Police Car

Green Race Car

Green Police Car

Red Race Car

Red Police Car

Race

Police

Race

Police

Race

Police

PaintCar

Multiplication Rule of CountingThe Multiplication Rule of Counting

applies to this type of situation If a task consists of a sequence of

choices With p selections for the first choice With q selections for the second

choice With r selections for the third choice …

Then the number of different tasks is

p • q • r • …

Example● Example Part A● A child is coloring a picture of a

shirt and pants● There are 5 different colors of

markers● How many ways can this be

colored?● By the multiplication rule5 • 5 = 25

Different Example● Example Part B● A child is coloring a picture of a shirt and

pants● There are 5 different colors of markers● The child wants to use 2 different colors● How many ways can this be colored?● By the multiplication rule5 • 4 = 20

Example continued

● Allowing the same marker to be used twice5 • 5 = 25● Requiring that there be two different

markers5 • 4 = 20● There are 5 selections for the first choice for

both Part A and Part B of this example● But they differ for the second choice … there

are only 4 selections for Part B

Repetition??

● Example continued● Part A, allowing the same marker to be

used twice, is called counting with repetition and has formulas such as

5 • 5 • 5 • …● Part B, requiring that there be two

different markers, is called counting without repetition and has formulas such as

5 • 4 • 3 • …

Calculator Commands While in a

CALCULATOR Page:

Menu, Probability,

We will be using Factorial, permutations, & combinations

Factorial● One way to help write these products is using

the factorial symbol n!n! = n • (n-1) • (n-2) • … • 2 • 1● We start off by saying that0! = 1 and 1! = 1● For example5! = 5 • 4 • 3 • 2 • 1 = 120● Notice how 5! looks like the 5 • 4 • 3 from the

previous example

Permutation (Order Matters)● The problem of choosing one marker out

of 5 and then choosing a second marker out of the 4 remaining is an example of a permutation

● A permutation is an ordered arrangement, in which r different objects are chosen out of n different objects with repetition not allowed

● The number of ways is written nPr

Permutation Formula A mathematical way to write the formula for the

number of permutations is

This is a very convenient mathematical way to write a formula for nPr, but it is not a particularly efficient way to actually compute it

In particular, n! gets rapidly gets very large

Order● For some problems, the order of choice

does not matter● Order matters example

Choosing one person to be the president of a club and another to be the vice-president

Two different roles● Order does not matter example

Choosing two people to go to a meeting The same role

Combination (Order Does Not Matter)● When order does not matter, this is

called a combination● A combination is an

unordered arrangement, in which r different objects are chosen out of n different objects with repetition not allowed

● The number of ways is written nCr

Permutation vs. Combination Comparing the description of a permutation

with the description of a combination

The only difference is whether order matters

Combination Formula Because each combination corresponds

to r! permutations, the formula nCr for the number of combinations is

Example● If there are 8 researchers and 3 of them are to

be chosen to go to a meeting● A combination since order does not matter

● There are 56 different ways that this can be done

Permutation or Combination● Is a problem a permutation or a

combination?● One way to tell

Write down one possible solution (i.e. Roger, Rick, Randy)

Switch the order of two of the elements (i.e. Rick, Roger, Randy)

● Is this the same result? If no – this is a permutation – order matters If yes – this is a combination – order does not

matter

DifferentOur permutation and combination

problems so far assume that all n total items are different

Sometimes we have a permutations but not all of the n items are different

This is a more complicated problemHow many ways are there?

Example● How many ways to put 3 A’s, 2 N’s, and 2

T’s to try to make a seven letter sequence?

____ ____ ____ ____ ____ ____ ____● Each of the blanks can be filled in with

either an A or a N or a T● The three A’s are the same … the two N’s

are the same … the two T’s are the same

Example ContinuedWhere can the A’s go?

There are 7 possible placesAny 3 of them are possibleOrder does not matterSo 7C3 different ways to put in the A’s

Example ContinuedWhere can the N’s go?

There are 4 possible places (since 3 of the 7 have been taken by the A’s already)

Any 2 of them are possible Order does not matter So 4C2 different ways to put in the N’s

And there are 2C2 different ways to put in the T’s

Example Continued● Altogether there are

7C3 • 4C2 • 2C2

different ways● This is

● Notice that the denominator is 3, 2, 2 … the numbers of each letter

Permutation● A permutation example● In a horse racing “Trifecta”, a gambler

must pick which horse comes in first, which second, and which third

● If there are 8 horses in the race, and every order of finish is equally likely, what is the chance that any ticket is a winning ticket?

● Order matters, so this is a permutations problem

Permutation Cont.A permutation example continuedThere are 8P3 permutations of the

order of finish of the horsesThe probability that any one ticket

is a winning ticket is 1 out of 8P3, or 1 out of 336

Combination Example● A combination example● The Powerball lottery consists of choosing 5

numbers out of 55 and then 1 number out of 42

● The grand prize is given out when all 6 numbers are correct

● What is the chance of getting the grand prize?

● Order does not matter, so this is a combinations problem (for the 5 balls)

Combination Cont.● A combination example continued● There are 55C5 combinations of the 5 numbers

● There are 42 possibilities for the last ball, so the probability of the grand prize is 1 out of

which is pretty small

SummaryThe Multiplication Rule counts the

number of possible sequences of itemsPermutations and combinations count

the number of ways of arranging items, with permutations when the order matters and combinations when the order does not matter

Permutations and combinations are used to compute probabilities in the classical method