Chapter 9 - Probability I

Front 1

A ________ of a set of objects is an ordering of the objects in a row.

Back 1

permutation

For example: the set of elements a, b, and c has six permutations:

abc, acb, cba, bac, bca, cab

Front 2

A permutation of a set of objects is an _______ of the objects in a row.

Back 2

ordering of the objects

For example: the set of elements a, b, and c has six permutations:

abc, acb, cba, bac, bca, cab

Front 3

For any integer n with n ≥ 1, the number of permutations of a set with nelements is ____

Back 3

n!

Front 4

For any integer n with n ≥ 1, the number of ________ of a set with n elements is n!

Back 4

permutations (ordering of the values)

Front 5

Permutations of objects around a circle:

At a meeting of diplomats, the six participants are to be seated around a circular table. It doesn't matter who sits in which chair, but it does matter how the diplomats are seated relative to each other. How many different ways can the diplomats be seated?

Back 5

Start by sitting one of the diplomats at any seat.

The other diplomats can be arranged in seats around that diplomat in all possible orders. So there are:

(6 - 1)! = 5! = 120 ways to seat the group.

Front 6

Given the set {a, b, c}, there are six ways to select two letters from the set and write them in order:

ab ac ba bc ca cb

Each such ordering of two elements of {a, b, c} is called a _______________ of {a, b, c}

Back 6

called a 2-permutation of {a, b, c}

Front 7

A _____________ of a set of n elements is an ordered selection of r elements taken from the set of n elements.

Back 7

r-permutation

The number of r-permutations of a set of n elements is denoted P(n, r)

Front 8

0! = ?

1! = ?

Back 8

0! = 1

1! = 1

Front 9

Formula for evaluating r-permutations:

P(n, r) = ?

Back 9

Front 10

P(n, 2) = ?

Back 10

Front 11

P(n, 1) = ?

Back 11

Front 12

The number of subsets of size r (r-combinations)that can be chosen from a set of n elements.

It is indicated by the symbol:

Back 12

which is read

"n choose r"

Front 13

The number of _____ of size r

(r-combinations)

that can be chosen from a set of n elements.

Back 13

the number of subsets of size r that can be chosen from a set of n elements

Front 14

n represents...

Back 14

n is the original set

"n choose r"

The number of subsets of size r that can be chosen from a set of n elements

Front 15

r represents...

Back 15

r is the size of the subsets

"n choose r"

The number of subsets of size r that can be chosen from a set of n elements

Front 16

The number of subsets of size r that can be chosen from a set S equals the number of subsetsof size r that S has. Each individual subset of size r is called an _-_____________ of the set.

Back 16

Each individual subset of size r is called an

r-combination of the set

Front 17

When the order doesn't matter, it is a _________.

When the order does matter it is a Permutation.

Back 17

No particular order:

Combination

Front 18

When the order doesn't matter, it is a Combination.

When the order does matter it is a __________.

Back 18

Order does matter

Permutation

Front 19

Suppose five members of a group of twelve are to be chosen to work as a teamon a special project. How many distinct five-person teams can be selected?

This is an example of a ___________. (combination or permutation)?

Back 19

Combination, the order does not matter.

Front 20

Let S = {Ann, Bob, Cyd, Dan}.

Each committee consisting of three of the four people inS is a 3-combination of S.

Back 20

Front 21

There are two distinct methods that can be used to select r objects from a set of n elements.

In an ordered selection, it is not only what elements are chosen but also theorder in which they are chosen that matters.

For this we use: (combination, permutation)?

Back 21

r-permutation

r is the size of the selection of n elements.

Front 22

There are two distinct methods that can be used to select r objects from a set of n elements.

In an unordered selection, on the other hand, it is only the identity of the chosen elements that matters.

For this we use: (combination, permutation)?

Back 22

r-combination

r is the size of the selection of n elements.

Front 23

There are two distinct methods that can be used to select r objects from a set of n elements.

In an __________ selection, it is not only what elements are chosen but also the order in which they are chosen that matters.

Back 23

ordered selection (permutations)

Front 24

There are two distinct methods that can be used to select r objects from a set of n elements.

In an ____________ selection, on the other hand, it is only the identity of the chosen elements that matters.

Back 24

unordered selection (combinations)

Front 25

An unordered selection of two elements from {0, 1, 2, 3} is the same as a 2-combination, or subset of size 2, taken from the set.

These are:

Back 25

6 total subsets

Front 26

The number of subsets of size r (or r-combinations) that can be chosen from a setof n elements, is given by the formula:

Back 26

(Add an r! to the bottom from the permutations equation)

Front 27

Back 27

Front 28

The number of distinct five-person teams (the number of subsets ofsize 5, [5combinations], that can be chosen from the set of twelve.) This equals:

Back 28

Front 29

Suppose two members of a group of twelve insist on working as a pair—any team mustcontain either both or neither.

How many five-person teams can be formed?

(Show using comb/perm, no need to calculate)

Back 29

Front 30

Suppose two members of the group don’t get along and refuse to work together on a team.

How many five-person teams can be formed?

(Show using comb/perm, no need to calculate)

Back 30

Front 31

The phrase at least n means “_________.”

Back 31

The phrase at least n means “n or more.”

For instance, if a set consists of three elements and you are to "choose at least two",

you will choose two or three.

Front 32

The phrase at most n means “__________.”

Back 32

The phrase at most n means “n or fewer.”

For instance, if a set consists of three elements and you are to "choose at most two",

you will choose none, one, or two

Front 33

How to solve:

Suppose the group of twelve consists of five men and seven women.

How many five-person teams can be chosen that consist of three men and two women?

Back 33

Step 1: Choose the men

Step 2: Choose the women

Step 3: Multiply those results together

Front 34

“How do I know what to multiply and what to add? When do I use the multiplication rule and when do I use the addition rule?” Unfortunately, these questions have no easy answers.

Starting advice:

Back 34

Imagine, as vividly as possible, the objects you are to count. You might even start to make an actual list of the items you are trying to count to get a sense for how to obtain them in a systematic way. You should then construct a model that would allow you to continue counting the objects one by one if you had enough time.

Front 35

“How do I know what to multiply and what to add? When do I use the multiplication rule and when do I use the addition rule?”

If you can imagine the elements to be counted as being obtained through a multistep process (in which each step is performed in a fixed number of ways regardless of how preceding steps were performed), then you can use the _________ rule.

Back 35

multiplication rule

The total number of elements will be the product of the number ofways to perform each step.

Front 36

“How do I know what to multiply and what to add? When do I use the multiplication rule and when do I use the addition rule?”

If you can imagine the set of elements to be counted as being broken up into disjoint subsets, then you can use the _______ rule.

Back 36

addition rule

The total number of elements in the set will be the sum of the number of elements in each subset.

Front 37

Suppose a collection consists of n objects of which:

n1 are of type 1 and are indistinguishable from each other

n2 are of type 2 and are indistinguishable from each other . . .

nk are of type k and are indistinguishable from each other,

The number of permutations of n is:

Back 37

Front 38

One of the most common mistakes students make in counting is...

Back 38

...to count certain possibilities more than once.

Front 39

Back 39

Front 40

The number of ways a set of size n can be partitioned into r subsets

can be written:

Back 40

The numbers Sn,r are called Stirling numbers of the second kind

Front 41

Back 41

Where:

Sn,1 = 1 and

Sn,n = 1

(initial conditions)

Front 42

The number of r-combinations with repetition allowed that can be selected from a set of n elements:

Back 42

Front 43

How many solutions are there to the equation x1+ x2+ x3+ x4= 10 ?

Think of the number 10 as divided into ten individual units and the variables x1, x2, x3, and x4 as four categories into which these units are placed.

Back 43

There are as many solutions to the equation as there are strings of ten crosses and three vertical bars

Front 44

How many integer solutions are there to the equation x1+ x2+ x3+ x4= 10

if eachxi≥1?

Hint: In this case imagine starting by putting one cross in each of the four categories.

Back 44

Then distribute the remaining six crosses among the categories. Such a distribution can be represented by a string of three vertical bars and six crosses.

Front 45

Which formula to use?

Ask both questions!!

1. __________________

2. __________________

Back 45

Two important questions of counting!

1. does order matter?

2. Is repetition allowed?

Front 46

Which formula to use?

1. Order DOES matter.

2. repetition IS allowed.

Back 46

1. Order DOES matter

2. repetition IS allowed

Front 47

Which formula to use?

1. Order does NOT matter.

2. repetition IS allowed.

Back 47

1. Order does NOT matter

2. repetition IS allowed

Front 48

Which formula to use?

1. Order DOES matter.

2. repetition is NOT allowed.

Back 48

1. Order DOES matter.

2. repetition is NOT allowed.

Front 49

Which formula to use?

1. Order does NOT matter.

2. repetition is NOT allowed.

Back 49

1. Order does NOT matter.

2. repetition is NOT allowed.