Table of Contents
How do you write a combinatorial argument?
In general, to give a combinatorial proof for a binomial identity, say A=B you do the following:
- Find a counting problem you will be able to answer in two ways.
- Explain why one answer to the counting problem is A.
- Explain why the other answer to the counting problem is B.
What is a combinatorial explanation?
Definition: A combinatorial interpretation of a numerical quantity is a set of combinatorial objects that is counted by the quantity. You find a set of objects that can be interpreted as a combinatorial interpretation of both the left hand side (LHS) and the right hand side (RHS) of the equation.
What is double counting in discrete mathematics?
In combinatorics, double counting, also called counting in two ways, is a combinatorial proof technique for showing that two expressions are equal by demonstrating that they are two ways of counting the size of one set. Since both expressions equal the size of the same set, they equal each other.
How do I prove my hockey stick identity?
The hockey stick identity gets its name by how it is represented in Pascal’s triangle. In Pascal’s triangle, the sum of the elements in a diagonal line starting with 1 is equal to the next element down diagonally in the opposite direction. Circling these elements creates a “hockey stick” shape: 1 + 3 + 6 + 10 = 20.
What is a counting argument?
Counting arguments are among the most basic proof methods in mathematics. A counting argument (in the context of formal methods) is a pro- gram proof that makes use of one or more counters, which are not part of the program itself, but which are useful for abstracting pro- gram behaviour.
What is counting argument?
Typically a “counting argument” refers to listing elements of a set in a meaningful way to show that the set is the the same size as the natural numbers or not the same size as the natural numbers.
What do you mean by error of double counting?
Double counting in accounting is an error whereby a transaction is counted more than once, for whatever reason. But in social accounting it also refers to a conceptual problem in social accounting practice, when the attempt is made to estimate the new value added by Gross Output, or the value of total investments.
What is Lucas theorem and how do you apply it?
In number theory, Lucas’s theorem expresses the remainder of division of the binomial coefficient. by a prime number p in terms of the base p expansions of the integers m and n. Lucas’s theorem first appeared in 1878 in papers by Édouard Lucas.
What is the hockey stick pattern?
A hockey stick chart is a chart characterized by a sharp increase after a relatively flat and quiet period. It is generally observed in scientific research measuring medical results or environmental studies. In cases of business sales, a hockey stick chart is represented by a sudden and dramatic increase in sales.
What is the principle of double counting?
The principle of double counting is to first count a set of objects one way, then count another way. This is best explained by looking at several examples, since there are many different ways to count.