Counting (Discrete Structures - Chapter 6) Flashcards
What is combinatorics?
The study of arrangements of objects
It is an important part of discrete mathematics.
When did the study of combinatorial questions arise?
In the seventeenth century
This was during the study of gambling games.
What is enumeration in combinatorics?
The counting of objects with certain properties
It is crucial for solving various problems.
Why is counting important in combinatorics?
To solve many different types of problems
It helps determine algorithm complexity and resource availability.
What are some applications of counting in modern contexts?
Determining telephone numbers and Internet protocol addresses
It also plays a key role in mathematical biology, especially in sequencing DNA.
What role do counting techniques play in probability?
They are used extensively when probabilities of events are computed
Counting is fundamental to understanding event likelihood.
What is the pigeonhole principle?
States that if there are more objects than boxes, at least one box contains at least two objects
This principle can be applied in various counting scenarios.
Give an example of the pigeonhole principle.
Among a set of 15 or more students, at least 3 were born on the same day of the week
This shows how the principle can be used in real-life scenarios.
What are permutations and combinations?
Ordered and unordered arrangements of objects
They are used in many counting problems.
What is a counting problem involving competitive exams?
Counting possible sets of 100 students invited to a banquet
This involves the top finishers of an exam taken by 2000 students.
What is another problem in combinatorics?
Generating all the arrangements of a specified kind
This is often important in computer simulations.
What will be studied in Section 6.1?
The basic rules of counting
These rules can solve a tremendous variety of problems.
What will be studied in Section 6.2?
The pigeonhole principle
This concept is a crucial combinatorial tool.
Fill in the blank: Counting is used to determine the _______ of algorithms.
complexity
Counting helps in assessing how algorithms perform.
True or False: Combinatorics has applications in mathematical biology.
True
It plays a key role in sequencing DNA.
Fill in the blank: Arrangements of objects can be made with or without _______.
repetitions
This distinction is important in counting problems.
