@MISC{_combinatoricscounts,

author = {},

title = {Combinatorics Counts TEXTBOOK UNIT OBJECTIVES},

year = {}

}

• Combinatorics is about organization. • Many combinatorial problems involve ways to enumerate, or count, various things in an efficient manner. • The counting function C(n,k), is a powerful tool used to count subsets of a larger set, or give coefficients in binomial expansions. • Bijection—the identification of a “one-to-one ” correspondence—enables us to enumerate a set that may be difficult to count in terms of another set that is more easily counted. • Pascal’s Triangle is an elegant illustration of the counting function C(n,k). • Techniques from graph theory can help with combinatorial challenges such as finding circular permutations. • The pigeonhole principle—the idea that if you have more pigeons than holes, some holes must have more than one pigeo—is a deceptively simple idea that can be used to prove startling results. • Ramsey Theory explains why we sometimes find order in supposed randomness. • Ideas from combinatorics are at play in modern methods of DNA sequencing. • The question of whether or not P = NP—whether certain types of seemingly computationally intractable combinatorial problems can be solved in reasonable amounts of time—is at the forefront of current research in both combinatorics and computer science. Mathematics may be defined as the economy of counting. There is no problem in the whole of mathematics which cannot be solved by direct counting.

combinatorics count textbook unit objective counting function reasonable amount certain type startling result various thing combinatorial challenge current research intractable combinatorial problem modern method computer science graph theory pigeonhole principle efficient manner binomial expansion supposed randomness ramsey theory one-to-one correspondence powerful tool simple idea many combinatorial problem pascal triangle elegant illustration direct counting circular permutation

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