04/19/23 MATH 106, Section 3 1

Section 3

Recipes

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04/19/23 MATH 106, Section 3 2

Let’s revisit our “tree” …

• You have 4 shirts (red, white, blue, and khaki) and 3 pairs of pants (khaki, black, and blue jeans). Construct a tree diagram to find how many different outfits can you make.

red

white

blue

khaki

khaki

black

blue jeans

khaki

black

blue jeans

khaki

black

blue jeans

khaki

black

blue jeans

shirts pants

The number of different outfits is 12.

04/19/23 MATH 106, Section 3 3

Can we come up with the number 12 in another way?• Notice that the number of outfits is

the number of shirts times the number of pants.

• That’s because multiplication is just a fast way to repeatedly add a number to itself.

• This leads us to …Simple Multiplication Principle: Consider a

task which can be done in two stages. If there are m ways to do the first step and n ways to do the second step, then the entire task can be done in m n ways.

red

white

blue

khaki

khaki

black

blue jeans

khaki

black

blue jeans

khaki

black

blue jeans

khaki

black

blue jeans

shirts pants

04/19/23 MATH 106, Section 3 4

Let’s look at the first example on the handout …

At lunch, kindergartners are allowed to pick one sandwich (from peanut butter, hamburger, or grilled cheese) and one dessert (from apple, banana, cake, ice cream, and cookies). How many possible combinations could they pick?

35 = 15 combinations

(page 18 of the textbook)

• This leads to the “General” Multiplication Principle …

#1

How many possible combinations could the children pick if they can also select a drink as well (from milk, juice, or water)?

04/19/23 MATH 106, Section 3 5

The Multiplication Principle• Multiplication Principle: Consider a task that can be accomplished in

k stages such that at each stage the number of choices does not depend on the previous choices made. Then the total number of ways to accomplish the entire task is equal to the product of the number of ways to do each of the k stages.

• NOTE: The number of choices at any stage must not depend on the previous choices!

353 = 45 combinations

(page 18 of the textbook)

How many possible combinations could the children pick if they can also select a drink as well (from milk, juice, or water)?

04/19/23 MATH 106, Section 3 6

The Multiplication Principle• Multiplication Principle: Consider a task that can be accomplished in

k stages such that at each stage the number of choices does not depend on the previous choices made. Then the total number of ways to accomplish the entire task is equal to the product of the number of ways to do each of the k stages.

• NOTE: The number of choices at any stage must not depend on the previous choices!

252 = 20 classifications

(page 18 of the textbook)

A study will classify people by sex (M, F), age (0-19, 20-39, 40-59, 60-79, 80+), and whether or not they own a computer. How many different classifications will there be?

#2

04/19/23 MATH 106, Section 3 7

What if latter choices depend on earlier choices?

You have 4 shirts (red, white, blue, and khaki) and 3 pairs of pants (khaki, black, and blue jeans). How many different outfits can you make if you don’t have the same color shirt and pants? To find the answer, first answer the following questions:

How many outfits do I have if I pick the red shirt?

How many if I pick the white shirt?

The blue shirt?

The khaki shirt?

3

This leads us to the Addition Principle …

The total number of possible outfits is the sum of these four numbers.

3

2

2

10

#3

04/19/23 MATH 106, Section 3 8

The Addition Principle• Addition Principle: Consider a task that can be accomplished in

k completely separate ways. Then the total number of ways to accomplish the entire task is equal to the sum of the number of ways to accomplish each of the k cases.

• We need to make sure that we get all of the cases!(page 20 of the textbook)

Five friends (Cass, Marg, Katie, Adrienne, and Camille) are going to the movies.

How many different seating arrangements are there if Cass and Adrienne insist on sitting on the ends?

RECIPE: Choose the seat for Cass and then choose the seat for Adrienne and then choose the seat for Marg and then choose the seat for Katie and then choose the seat for Camille.

(2)(1)(3)(2)(1) = 12

#4

04/19/23 MATH 106, Section 3 9

Five friends (Cass, Marg, Katie, Adrienne, and Camille) are going to the movies.

How many different seating arrangements are there if Cass and Adrienne insist on sitting on the ends?

RECIPE: Choose the seat for Cass and then choose the seat for Adrienne and then choose the seat for Marg and then choose the seat for Katie and then choose the seat for Camille. (2)(1)(3)(2)(1) = 12

RECIPE: Choose the two seats for Marg and Katie and then choose the seat for Marg and then choose the seat for Katie and then choose the seat for Cass and then choose the seat for Adrienne and then choose the seat for Camille. (4)(2)(1)(3)(2)(1) =

48

How many different seating arrangements are there if the only restriction is that Marg and Katie insist on sitting next to each other?

#4

04/19/23 MATH 106, Section 3 10

Using recipes …• The easiest way to apply the Multiplication Principle and the

Addition Principle is to write a “recipe” for creating one of the objects to be counted.

• A recipe is a list of directions that, if followed exactly, will build one possible example.

• An important requirement of a good recipe is that every possible result can be obtained from it (by the appropriate choices) and no result can be obtained in more than one way.

• The Multiplication Principle tells us that every time we have an “and then” we can multiply.

• The Addition Principle tells us that every time we have an “or” we can add.

04/19/23 MATH 106, Section 3 11

At lunch, kindergartners are allowed to pick one sandwich (from peanut butter, hamburger, or grilled cheese) and one dessert (from apple, banana, cake, ice cream, and cookies). How many possible combinations could they pick?

How many possible combinations could the children pick if they can also select a drink as well (from milk, juice, or water)?

RECIPE: Choose one sandwich and then choose one dessert.

RECIPE: Choose one sandwich and then choose one dessert and then choose one drink.

#1

04/19/23 MATH 106, Section 3 12

You have 4 shirts (red, white, blue, and khaki) and 3 pairs of pants (khaki, black, and blue jeans). How many different outfits can you make if you don’t have the same color shirt and pants?RECIPE: Choose one shirt and then choose one pair of pants. Why will this not give the correct answer?

RECIPE:Choose the red shirt and then choose one pair of pants

orchoose the white shirt and then choose one pair of pants

orchoose the blue shirt and then choose one pair of pants other than the blue jeans

orchoose the khaki shirt and then choose one pair of pants other than the khaki pants.

#3

04/19/23 MATH 106, Section 3 13

How many 4 letter “words” (not real words, just combinations of letters) are possible

if letters can be repeated?

if letters can be repeated and the last letter must be one of the five basic vowels?

if letters can be repeated and the last letter must not be one of the five basic vowels?

if letters can be repeated and the last letter must be e?

RECIPE: Choose the first letter and then choose the second letter and then choose the third letter and then choose the last letter.

(26)(26)(26)(26) = 456976

(26)(26)(26)(5) = 87880

(26)(26)(26)(21) = 369096

#5

Complete the handout problems we do not finish here in class as part ofthe homework for next class…

04/19/23 MATH 106, Section 3 14

How many 4 letter “words” (not real words, just combinations of letters) are possible

if letters can be repeated?

if letters can be repeated and the last letter must be one of the five basic vowels?

if letters can be repeated and the last letter must not be one of the five basic vowels?

if letters can be repeated and the last letter must be e?

if letters can not be repeated?

RECIPE: Choose the first letter and then choose the second letter and then choose the third letter and then choose the last letter.

(26)(26)(26)(26) = 456976

(26)(26)(26)(5) = 87880

(26)(26)(26)(21) = 369096

(26)(26)(26)(1) = 17576

(26)(25)(24)(23) = 358800

04/19/23 MATH 106, Section 3 15

There are 10 men and 12 women in the class. Select a pair of students of the same sex.

RECIPE: Choose a pair of men or choose a pair of women.

Select one man and then select a different man.

Select a woman and then select a different woman.

(10)(9) + (12)(11)What correction must be made?!?

——— + 2

——— = 111 2

Our recipe is selecting a pair of students in order, but we do not care about order! In other words, selecting Jean and then Joan gives us the same pair as selecting Joan and then Jean.

#6

04/19/23 MATH 106, Section 3 1604/19/23 MATH 106, Section 2 16

Homework Hints:In Section 3 Homework Problem #4,

In Section 3 Homework Problem #7,

In Section 3 Homework Problem #8,

In Section 3 Homework Problem #9,

notice how similar this to #6 on the class handout for Section #3.

note that you must choose a roll, a meat, and a cheese; but you may choose lettuce or no lettuce.

don’t be confused or fooled by the way the sentence is worded. Consider “cream” possibilities and “sugar” possibilities separately.

consider each individual choice separately as a “yes” or “no”.