Counting MethodsCHAPTER 2

Chapter 22.1 – COUNTING

PRINCIPLES

David lives in Winnipeg. This summer he plans a sight-seeing trip that includes visiting his family in Regina and Saskatoon. He has chosen and mapped out three different routes he can take from Winnipeg to Regina and two different routes he can take from Regina to Saskatoon.

How many different routes could David take to get from Winnipeg to Saskatoon?

EXAMPLE

Hannah plays on her school soccer team. The soccer uniform has:• Three different sweaters: red, white, and black, and• Three different shorts: red, white, and black.

How many different variations of the soccer uniform can the coach choose from for each game?

Use a tree diagram: Or, use the Fundamental Counting Principle, which states that if there are a ways to perform one task and b ways to perform another, then there are a × b ways of performing both.

If U is the number of uniform variations:

U = (number of sweaters) x (number of shorts) U = 3 x 3 = 9

EXAMPLE

A luggage lock opens with the correct three-digit code. Each wheel rotates through the digits 0 to 9. a) How many different three-digit codes are possible?b) Suppose each digit can be used only once in a code. How many

different codes are possible when repetition is not allowed?

a) Using the Counting Principle, the number of different codes, C, is:

C = D1 x D2 x D3

C = 10 x 10 x 10C = 1000

There are a 1000 different possible codes.

b) If there are no repeats available, then how many different digits can D1 be? How about D2? How about D3?

C = D1 x D2 x D3

C = 10 x 9 x 8C = 720

If repeats are not allowed, then there are 720 possible different codes.

EXAMPLE

A standard deck of cards contains 52 cards. Count the number of possibilities of drawing a single card and getting:a) either a black face card or an aceb) either a red card or a 10

a) Event A = {drawing a black face card}

Event B = {drawing an ace}

n(A B) = n(A) + n(B)

How many black face cards are there? 6How many aces are there? 4

n(A B) = 6 + 4 = 10There are 10 ways.

b) Try it!

Independent Practice

PG. 73-75, # 1, 2, 5, 9, 11, 15.

Chapter 2

2.2 – INTRODUCING PERMUTATIONS AND

FACTORIAL NOTATION

EXAMPLE

Naomi volunteers at an after-school daycare. At around 4 pm, she lines up her group of children at the fountain to get a drink of water. Determine the number of arrangements that six children can form while lining up to drink.

This is called a permutation, which is an arrangement of distinguish-able objects in a definite order.

EXAMPLE

Evaluate the following.

a) b)

a) 10! = 10 x 9 x 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1 10! = 3 628 800

b)

EXAMPLE

Simplify, where n N.

a) b)

EXAMPLE

Solve:

Independent Practice

PG. 81-83, # 1, 2, 5, 6, 8, 11, 13.

Chapter 2

2.3 – PERMUTATIONS WHEN ALL OBJECTS

ARE DISTINGUISHABLE

PERMUTATIONS

When dealing with permutations, we use the notation nPr to represent the number of permutations that can be made from a set of n different objects where only r of them are used in each arrangement.

Formula:

For example, if there are five names on a ballot for an election, and you’re voting on 3 different positions then:

n = 5 r = 3 So we’d be finding 5P3

EXAMPLE

Matt has downloaded 10 new songs from an online music store. He wants to create a playlist using 6 of these songs arranged in any order. How many different 6-song playlists can be created from his new downloaded songs?

n is the number of objects to choose from. In this case, what’s n?

r is the number of objects used in the arrangement. In this case, what’s r?

n = 10

r = 6

There are 151 200 differ-ent 6-song arrangements.

SOME RULES/CALCULATOR

O! = 1

Calculator:

Press whatever number is n MATH Arrow left over to PRB 2: nPr Press the number for r ENTER

EXAMPLE

Tania needs to create a password for a social networking website she registered with. The password can use any digits from 0 to 9 and/or any letters of the alphabet. The password in case sensitive, so she can use both lower- and upper-case letters. A password must be at least 5 characters to a maximum of 7 characters, and each character can be used only once in the password. How many different passwords are possible?

How many different characters can you use?Number of characters = 10 digits + 26 lower-case letters + 26

upper-case letters Number of characters = 62

What’s r in this case? (How many characters can the password be?)Case 1: r = 5Case 2: r = 6Case 3: r = 7

So, the total number of passwords is all of these cases added together.

EXAMPLE

At a used car lot, seven different car models are to be parked close to the street for easy viewing.a) The three red cars must be parked so that there is a red car at each end

and the third red car is exactly in the middle. How many ways can they be parked?

b) The three red cars must be parked side by side. How many ways can they be parked?a) If we call the positions P1 – P7,

what positions must the red cars be in?

So, it ends up being:

The cars can be parked 144 ways.

b) The red cars need to be side-by-side… So how many different places can they go?

The cars can be parked 720 ways.

Independent Practice

PG. 93-94, # 1, 3, 4, 5, 7, 9, 11, 15.

Chapter 2

2.4 – PERMUTATIONS WHEN OBJECTS ARE

IDENTICAL

EXAMPLE

In the mountainous regions of India, China, Nepal, and Bhutan, it is common to see prayer flags. Each flag has a prayer written on it, and colour is used to symbolize different elements: green (water), yellow (earth), white (air/wind), blue (sky/space), and red (fire).

How many different arrangements of the same prayer can Dorji make using these 9 flags: 1 green, 1 yellow, 2 white, 3 blue, and 2 red?

When permutations involve identical objects, then for n objects, and a, b, c, etc… numbers of identical objects, we use the formula:

There are 15 120 different prayer flag arrangements.

EXAMPLE

How many ways can the letters of the word CANADA be arranged, if the first letter must be N and the last letter must be C?

How many letters are left to be arranged, if the first and last letters are decided?

There are four letters left to arrange, so that means that n = 4.

How many identical letters do we have? 3 a’s

The letters of CANADA can be arranged in 4 different ways, if the first letter is N and the last is C.

EXAMPLE

Julie’s home is three blocks north and five blocks west of her school. How many routes can Julie take from home to school if she always travels either south or east?

Independent Practice

PG. 104-107 #1, 4, 6, 7, 9, 11, 13.

Chapter 22.6 - COMBINATIONS

COMBINATIONS

When the order of an arrangement doesn’t matter, we are dealing with combinations. For instance, if you were grouping together students, it wouldn’t matter if you grouped them as “Paul, Susan, and Jack,” “Paul, Jack, and Susan,” “Susan, Paul, and Jack,” “Susan, Jack, and Paul,” “Jack, Susan, and Paul,” or “Jack, Paul, and Susan.” They are six different permutations of the three names, but they all represent just one combination.

In any given situation, will there be more permutations or more combinations?

EXAMPLE

Each year during the Festival du Voyageur, held during February in Winnipeg, Manitoba, high schools compete in the Voyageur Snow Sculpture Contest. This year Amir’s school will enter a three-person team. Nine students have volunteered to be on the team.

Determine the number of three-person teams that can be formed from the 9 volunteers.

Combinations Notation:

or

n = 9r = 3

The number of three-person teams that can be formed is 84.

EXAMPLE

A restaurant serves 10 flavours of ice cream. Danielle has ordered a large sundae with three scoops of ice cream. How many different ice cream combinations does Danielle have to choose from, if she wants each scoop to be a different flavour?

Does the order of the flavours matter?

n = 10r = 3

There are 120 different combinations of ice cream flavours that she could choose.

EXAMPLE

A planning committee is to be formed for a school-wide Earth Day program. There are 13 volunteers: 8 teachers and 5 students. How many ways can the principal choose a 4-person committee that has at least 1 teacher?

How many teachers can there be in the 4-person committee?

4 cases

Case 1: 1 teacher, 3 students

Case 2: 2 teachers, 2 students

Both the teacher and the students are sep-arate combinations. We need to multiply them together.

Case 3: 3 teachers, 1 student

Case 4: 4 teachers, 0 students

710 committees

Independent practice

PG. 118-120, #2, 4, 6, 8, 10, 11, 18.

Chapter 2

2.7 – SOLVING COUNTING PROBLEMS

EXAMPLE

A piano teacher and her students are having a group photograph taken. There are three boys and five girls. The photographer wants the boys to sit together and the girls to sit together for one of the poses. How many ways can the students and teacher sit in a row of nine chairs for this pose?

The teachers, boys and girls are all sitting together. How many ways can the three groups be arranged?

3! = 3 x 2 x 1 = 6

Within each group:

How many ways can the teacher sit?

1

How many ways can the boys sit?

3! = 6

How many ways can the girls sit?

5! = 5 x 4 x 3 x 2 x 1 = 120

So how many ways are there in total?

6 x 1 x 6 x 120 = 4320

There are 4320 different ways for the

group to sit.

EXAMPLE

Combination problems are common in computer science. Suppose there is a set of 10 different data items represented by {a, b, c, d, e, f, g, h, i, j} to be placed into four different memory cells in a computer. Only 3 data items are to be placed in the first cell, 4 data items in the second cell, 2 data items in the third cell, and 1 data item in the last cell. How many ways can the 10 data items be placed in the four memory cells?

1st cell: 3 items out of 10

2nd cell:How many items left?

4 items out of 7

3rd cell:How

many items left?

2 items out of 3

4th cell:How many items left?

1 item out of 1

How do we find the total number of combinations?

120 x 35 x 3 x 1 = 12 600 12 600 combinations

EXAMPLE

How many different five-card hands that contain at most one black card can be dealt to one person from a standard deck of playing cards?

What are the possible numbers of black cards? Either 0 or 1, so there’s two cases

Case 1: 1 black card and 4 redsCase 2: 0 black cards and 5 reds

Does the order matter, here?

Case 1:

How many red/black cards are there?

Case 2:

The total number of possible hands is 388 700 + 65 780 = 454 480.

Independent practice

PG. 126-128, #1, 3, 4, 5, 8, 10, 12.