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- https://math.libretexts.org/Bookshelves/Combinatorics_and_Discrete_Mathematics/Discrete_Mathematics_(Levin)/1%3A_CountingOne of the first things you learn in mathematics is how to count. Now we want to count large collections of things quickly and precisely. For example: In a group of 10 people, if everyone shakes hands...One of the first things you learn in mathematics is how to count. Now we want to count large collections of things quickly and precisely. For example: In a group of 10 people, if everyone shakes hands with everyone else exactly once, how many handshakes took place? How many ways can you distribute 1010 girl scout cookies to 77 boy scouts? How many anagrams are there of “anagram”? Before tackling questions like these, let's look at the basics of counting.
- https://math.libretexts.org/Courses/Monroe_Community_College/Supplements_for_Discrete_Judy_Dean/02%3A_Counting/2.01%3A_Additive_and_Multiplicative_PrinciplesConsider this rather simple counting problem: at Red Dogs and Donuts, there are 14 varieties of donuts, and 16 types of hot dogs. If you want either a donut or a dog, how many options do you have? Thi...Consider this rather simple counting problem: at Red Dogs and Donuts, there are 14 varieties of donuts, and 16 types of hot dogs. If you want either a donut or a dog, how many options do you have? This isn't too hard, just add 14 and 16. Will that always work? What is important here?
- https://math.libretexts.org/Courses/Saint_Mary's_College_Notre_Dame_IN/SMC%3A_MATH_339_-_Discrete_Mathematics_(Rohatgi)/Text/5%3A_Graph_Theory/5.5%3A_Euler_Paths_and_CircuitsAn Euler path, in a graph or multigraph, is a walk through the graph which uses every edge exactly once. An Euler circuit is an Euler path which starts and stops at the same vertex. Our goal is to fin...An Euler path, in a graph or multigraph, is a walk through the graph which uses every edge exactly once. An Euler circuit is an Euler path which starts and stops at the same vertex. Our goal is to find a quick way to check whether a graph (or multigraph) has an Euler path or circuit.
- https://math.libretexts.org/Courses/SUNY_Schenectady_County_Community_College/Discrete_Structures/11%3A_Counting/11.04%3A_Combinatorial_ProofsTo give a combinatorial proof for a binomial identity, say A=B you do the following: (1) Find a counting problem you will be able to answer in two ways. (2) Explain why one answer to the counting pr...To give a combinatorial proof for a binomial identity, say A=B you do the following: (1) Find a counting problem you will be able to answer in two ways. (2) Explain why one answer to the counting problem is A. (3) Explain why the other answer to the counting problem is B. Since both A and B are the answers to the same question, we must have A=B. The tricky thing is coming up with the question.
- https://math.libretexts.org/Bookshelves/Combinatorics_and_Discrete_Mathematics/Discrete_Mathematics_(Levin)/1%3A_Counting/1.6%3A_Advanced_Counting_Using_PIEThe Principle of Inclusion/Exclusion (PIE) gives a method for finding the cardinality of the union of not necessarily disjoint sets. If there is any overlap among the sets, those elements are counted...The Principle of Inclusion/Exclusion (PIE) gives a method for finding the cardinality of the union of not necessarily disjoint sets. If there is any overlap among the sets, those elements are counted multiple times. So we subtract the things in each intersection of a pair of sets. But doing this removes elements which are in all three sets once too often, so we need to add it back in.
- https://math.libretexts.org/Courses/SUNY_Schenectady_County_Community_College/Discrete_Structures/11%3A_Counting/11.01%3A_Additive_and_Multiplicative_PrinciplesConsider this rather simple counting problem: at Red Dogs and Donuts, there are 14 varieties of donuts, and 16 types of hot dogs. If you want either a donut or a dog, how many options do you have? Thi...Consider this rather simple counting problem: at Red Dogs and Donuts, there are 14 varieties of donuts, and 16 types of hot dogs. If you want either a donut or a dog, how many options do you have? This isn't too hard, just add 14 and 16. Will that always work? What is important here?
- https://math.libretexts.org/Courses/Saint_Mary's_College_Notre_Dame_IN/SMC%3A_MATH_339_-_Discrete_Mathematics_(Rohatgi)/Text/3%3A_Symbolic_Logic_and_Proofs/3.3%3A_ProofsAnyone who doesn't believe there is creativity in mathematics clearly has not tried to write proofs. Finding a way to convince the world that a particular statement is necessarily true is a mighty und...Anyone who doesn't believe there is creativity in mathematics clearly has not tried to write proofs. Finding a way to convince the world that a particular statement is necessarily true is a mighty undertaking and can often be quite challenging. There is not a guaranteed path to success in the search for proofs. Writing proofs is a bit of an art. Like any art, to be truly great at it, you need some sort of inspiration, as well as some foundational technique.
- https://math.libretexts.org/Courses/Saint_Mary's_College_Notre_Dame_IN/SMC%3A_MATH_339_-_Discrete_Mathematics_(Rohatgi)/Text/2%3A_Sequences/2.1%3A_DefinitionsWhile we often just think of sequences as an ordered list of numbers, they really are a type of function. Later we will manipulate sequences in much the same way you have manipulated functions in alge...While we often just think of sequences as an ordered list of numbers, they really are a type of function. Later we will manipulate sequences in much the same way you have manipulated functions in algebra or calculus. We can shift a sequence up or down, add two sequences, or ask for the rate of change of a sequence. These are done exactly as you would for functions.
- https://math.libretexts.org/Courses/Monroe_Community_College/Supplements_for_Discrete_Judy_Dean/02%3A_Counting/2.05%3A_Stars_and_BarsMany of the counting problems in this section might at first appear to be examples of counting functions. After all, when we try to count the number of ways to distribute cookies to kids, we are assig...Many of the counting problems in this section might at first appear to be examples of counting functions. After all, when we try to count the number of ways to distribute cookies to kids, we are assigning each cookie to a kid, just like you assign elements of the domain of a function to elements in the codomain.
- https://math.libretexts.org/Courses/SUNY_Schenectady_County_Community_College/Discrete_Structures/11%3A_Counting/11.05%3A_Stars_and_BarsMany of the counting problems in this section might at first appear to be examples of counting functions. After all, when we try to count the number of ways to distribute cookies to kids, we are assig...Many of the counting problems in this section might at first appear to be examples of counting functions. After all, when we try to count the number of ways to distribute cookies to kids, we are assigning each cookie to a kid, just like you assign elements of the domain of a function to elements in the codomain.
- https://math.libretexts.org/Courses/Saint_Mary's_College_Notre_Dame_IN/SMC%3A_MATH_339_-_Discrete_Mathematics_(Rohatgi)/Text/5%3A_Graph_Theory/5.6%3A_Matching_in_Bipartite_GraphsGiven a bipartite graph, a matching is a subset of the edges for which every vertex belongs to exactly one of the edges. Our goal in this activity is to discover some criterion for when a bipartite gr...Given a bipartite graph, a matching is a subset of the edges for which every vertex belongs to exactly one of the edges. Our goal in this activity is to discover some criterion for when a bipartite graph has a matching.
