Counting

Question 1

(Use a tree diagram)

Answer 1

Question 2

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Question 3

Revision: Multiplciation rule

Question 4

Answer 4

12 times

Question 5

What is the defintion of a permutation?

Answer 5

An ordering of objects in a row. For example, the set {a, b, c} has 6 permutations:

Question 6

Given a set with n elements. What is the number of possible permutations?

Answer 6

n!

Question 7

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Question 8

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Question 9

Define what is meant by a 2-permutation of the set S with 9 elements

Answer 9

The ordering of 2 elements (In this case, its called pairing) from the set S. Denoted as P(n, r) - remember that n comes before r (like in the alphabet)
Example:

Question 10

What is the formula for permutations

Answer 10

Question 11

Just some extra info

Question 12

Question 13

What does 0! equal?

Answer 13

1

Question 14

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Question 15

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Question 16

Given a set A

What does N(A) mean?

Answer 16

The number of elements in A

Question 17

Just some info on the addition rule. (common sense)

N(A) means the number of elements in A

Question 18

How many three-digit integers (integers from 100 to 999 inclusive) are divisible by 5?

Use the addition rule.

Answer 18

Question 19

What is the difference rule?

Answer 19

Given two sets, A and B where the number of elements in each is known and B is a subset of A. The difference rule states that the N(A-B) = N(A)-N(B)

N() means the number of elements

Question 20

Consider a pin made from exactly four symboles chosen from 26 uppercase letters of the roman alphabet and the ten digits. There are 1,679,6166 pins with repition allows and 265,896 pins with no repeated symbol.
a) How many pins contain at least one repeated symbol?
b) If all pins are equally likely, what is the probability that a randomly chosen pin contains at least one repeated symbol?

Use addition or difference rule

Answer 20

Question 21

What is the probability of the complement of an event?

Answer 21

It means the probability of the event not happening.

The formula is just common sense

Question 22

Answer 22

Note that part b can be done by P(26, 3)+P(26, 4)+P(26, 5)

Question 23

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Question 24

Given a set S with n elements

Defined r-combination

Answer 24

An r-combination is is the number of subsets of S with each subset having r elements.

Question 25

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Question 26

# Given a set S Would you use to permutations or combinations to find the number of subsets of r elements (if order matters)

Answer 26

Permutations

Question 27

# Given a set S Would you use to permutations or combinations to find the number of subsets of r elements (if order doesnt matter)

Answer 27

Combinations

Question 28

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Question 29

What is the formula for combinations

Answer 29

Question 30

What is 0-combination for any set?

Answer 30

1

Question 31

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Question 32

Suppose a group you are to choose 5 members from a group of 12. But two members insist on working as a pair. And so any group must contain either the pair or not

Answer 32

Question 33

Given a group of 12 teams.

Answer 33

Question 34

Suppose a group of twelve consists of five men and seven women

Answer 34

Question 35

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Question 36

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Question 37

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Question 38

What is pascal's formula?

Answer 38

Sometimes refered to as pascal's triangle

Question 39

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Question 40

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Question 41

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Question 42

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Question 43

What is the Inclusion/Exlusion rule for two sets?

Answer 43

The following iamge is true for the following reason: N(A): N# elements in A and in common with B N(B): N# elements in A and in common with A And so when you add them, A∩B occurs twice, so you minus it to remove it

Question 44

What is the Inclusion/Exlusion rule for three sets?

Answer 44

Question 45

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Question 46

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Question 47

What is the pigeonhol principle in terms of functions?

Answer 47

A function that maps from a finite set to a smaller finite set cannot be one to one. | Atleast two elements must have the same image in the co-domain

Question 48

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Question 49

It is known that New York City has a population of at least 8,000,000 people and each person has at most 300,000 hairs on their head

Answer 49

Question 50

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Question 51

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Question 52

What is the pigeonhole principle in generalised terms?

Answer 52

If n pigeons fly into m pigeonholes and km > n for some integer k, then at least one pigeonhole must have k+1 pigeons

Question 53

Answer 53

Question 54

Use the genralised defintion of pigeonhole principle to define it with graphs

Answer 54