10.1 – Counting by Systematic Listing

One-Part Tasks

The results for simple, one-part tasks can often be listed easily. Tossing a fair coin:

Rolling a single fair die

Heads or tails

1, 2, 3, 4, 5, 6

Consider a club N with four members:

There are four possible results:

In how many ways can this group select a president?

M, A, T, and H.

N = {Mike, Adam, Ted, Helen} or N = {M, A, T, H}

10.1 – Counting by Systematic ListingProduct Tables for Two-Part Tasks

Determine the number of two-digit numbers that can be written using the digits from the set {2, 4, 6}.

The task consists of two parts:1. Choose a first digit2. Choose a second digit

The results for a two-part task can be pictured in a product table.

Second Digit

First Digit

246

2 4 622

4464

244262

264666

9 possible numbers

10.1 – Counting by Systematic ListingProduct Tables for Two-Part Tasks

What are the possible outcomes of rolling two fair die?

10.1 – Counting by Systematic ListingProduct Tables for Two-Part Tasks

Find the number of ways club N can elect a president and secretary.

The task consists of two parts:1. Choose a president 2. Choose a secretary

Secretary

M A T H

Pres. M

A

T

H

N = {Mike, Adam, Ted, Helen} or N = {M, A, T, H}

HHHM

TM

AM

MM

AH

MHMTMA

ATAA

THTTTA

HTHA

12 outcomes

10.1 – Counting by Systematic ListingProduct Tables for Two-Part Tasks

Find the number of ways club N can elect a two member committee.

Secretary

M A T H

Pres. M

A

T

H

N = {Mike, Adam, Ted, Helen} or N = {M, A, T, H}

HHHM

TM

AM

MM

AH

MHMTMA

ATAA

THTTTA

HTHA

6 committees

10.1 – Counting by Systematic ListingTree Diagrams for Multiple-Part Tasks

A task that has more than two parts is not easy to analyze with a product table. Another helpful device is a tree diagram.

Find the number of three digit numbers that can be written using the digits from the set {2, 4, 6} assuming repeated digits are not allowed.

A product table will not work for more than two digits.

Generating a list could be time consuming and disorganized.

10.1 – Counting by Systematic ListingTree Diagrams for Multiple-Part Tasks

Find the number of three digit numbers that can be written using the digits from the set {2, 4, 6} assuming repeated digits are not allowed.

26

6 246

1st # 2nd # 3rd #

4

6

4

2

6

2

4

4

6

2

4

2

264

426

462

624

642

6 possibilities

10.1 – Counting by Systematic ListingOther Systematic Listing Methods

There are additional systematic ways to produce complete listings of possible results besides product tables and tree diagrams.

A B

C

D

E

F

How many triangles (of any size) are in the figure below?

One systematic approach is begin with A, and proceed in alphabetical order to write all 3-letter combinations (like ABC, ABD, …), then cross out ones that are not triangles and those that repeat.

Another approach is to “chunk” the figure to smaller, more manageable figures.

There are 12 triangles.

10.2 – Using the Fundamental Counting PrincipleUniformity Criterion for Multiple-Part Tasks:

A multiple part task is said to satisfy the uniformity criterion if the number of choices for any particular part is the same no matter which choices were selected for previous parts.

Find the number of three letter combinations that can be written using the letters from the set {a, b, c} assuming repeated letters are not allowed.2 dimes and one six-sided die numbered from 1 to 6 are tossed. Generate a list of the possible outcomes by drawing a tree diagram.

A computer printer allows for optional settings with a panel of five on-off switches. Set up a tree diagram that will show how many setting are possible so that no two adjacent switches can be on?

Uniformity exists:

Uniformity does not exists:

Uniformity

ac

c abc

1st letter 2nd letter 3rd letter

b

c

b

a

c

a

b

b

c

a

b

a

acb

bac

bca

cab

cba

6 possibilities

10.2 – Using the Fundamental Counting Principle

Find the number of three letter combinations that can be written using the letters from the set {a, b, c} assuming repeated letters are not allowed.

Uniformity

1 d1

Die # Dime

12 possibilities

10.2 – Using the Fundamental Counting Principle

2 dimes and one six-sided die numbered from 1 to 6 are tossed. Generate a list of the possible outcomes by drawing a tree diagram.

2

4

6

1

3

5

d1

d2

d1

d2

d1

d2

d1

d2

d1

d2

d1

d2

1 d2

2 d1

2 d2 3 d1

3 d2 4 d1

4 d2 5 d1

5d2 6 d1

6 d2

Uniformity does not exist

1st switch 2nd switch 3rd switch

f

o

o

f

o

f

o

f

10.2 – Using the Fundamental Counting Principle

A computer printer is designed for optional settings with a panel of three on-off switches. Set up a tree diagram that will show how many setting are possible so that no two adjacent switches can be on? (o = on, f = off)

o

fo

f

o

f

10.2 – Using the Fundamental Counting PrincipleFundamental Counting Principle

The principle which states that all possible outcomes in a sample space can be found by multiplying the number of ways each event can occur.

Example:At a firehouse fundraiser dinner, one can choose from 2 proteins (beef and fish), 4 vegetables (beans, broccoli, carrots, and corn), and 2 breads (rolls and biscuits). How many different protein-vegetable-bread selections can she make for dinner?

Proteins Vegetables Breads

2 4 2 =

16 possible selections

10.2 – Using the Fundamental Counting PrincipleExample

At the local sub shop, customers have a choice of the following: 3 breads (white, wheat, rye), 4 meats (turkey, ham, chicken, bologna), 6 condiments (none, brown mustard, spicy mustard, honey mustard, ketchup, mayo), and 3 cheeses (none, Swiss, American). How many different sandwiches are possible?

Breads Meats Condiments Cheeses

3 4 6 =

216 possible sandwiches

3

10.2 – Using the Fundamental Counting PrincipleExample:

Consider the set of digits: {0, 1, 2, 3, 4, 5, 6, 7, 8, 9}.

(a) How many two digit numbers can be formed if repetitions are allowed?

(b) How many two digit numbers can be formed if no repetitions are allowed?

(c) How many three digit numbers can be formed if no repetitions are allowed?

1st digit 2nd digit

9 9 81 =1st digit 2nd digit

9 10 90 =

1st digit 2nd digit 3rd digit9 9 8 = 648

10.2 – Using the Fundamental Counting PrincipleExample:

(a) How many five-digit codes are possible if the first two digits are letters and the last three digits are numerical?

1st digit 2nd digit 3rd digit 4th digit 5th digit

676000 possible five-digit codes

26 26 10 1010

(a) How many five-digit codes are possible if the first two digits are letters and the last three digits are numerical and repeats are not permitted?

1st digit 2nd digit 3rd digit 4th digit 5th digit

468000 possible five-digit codes

26 25 10 89

10.2 – Using the Fundamental Counting PrincipleFactorials

For any counting number n, the product of all counting numbers from n down through 1 is called n factorial, and is denoted n!.

For any counting number n, the quantity n factorial is calculated by:

n! = n(n – 1)(n – 2)…(2)(1).

Examples:

a) 4! b) (4 – 1)! c)

4321 24

3!

321

654 20= =

Definition of Zero Factorial:

0! = 1

10.2 – Using the Fundamental Counting Principle

Example:

Arrangements of Objects

Factorials are used when finding the total number of ways to arrange a given number of distinct objects.

The total number of different ways to arrange n distinct objects is n!.

How many ways can you line up 6 different books on a shelf?

6 5 4 23 1

720 possible arrangements

10.2 – Using the Fundamental Counting Principle

Example:

9!

3! 2!30240 possible arrangements

Arrangements of n Objects Containing Look-Alikes

The number of distinguishable arrangements of n objects, where one or more subsets consist of look-alikes (say n1 are of one kind, n2 are of another kind, …, and nk are of yet another kind), is given by

1 2

!.

! ! !k

n

n n n

Determine the number of distinguishable arrangements of the letters of the word INITIALLY.

9 letters with 3 I’s and 2 L’s