23 - Introduction Permutations Discrete Mathematics Andrei...

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Introduction Permutations Discrete Mathematics Andrei Bulatov

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Discrete Mathematics - Permutations 23-2 What is Combinatorics Combinatorics, the study of arrangements of objects, is an important part of discrete mathematics. This subject was studied as long ago as the seventeenth century, when combinatorial questions arose in the study of gambling games. Enumeration is the counting of objects with certain properties Combinatorics is used in - Discrete probability: What is the probability to guess a 6-symbols password in the first attempt? - Analysis of algorithms: Why a comparison sort algorithm cannot be more efficient than O(n log n)? - Probabilistic proofs: Show that the local search algorithm with high probability does not find a good solution to a problem.
Discrete Mathematics - Permutations 23-3 The Rule of Sum If a first task can be performed in m ways, while a second task can be performed in n ways, and the two tasks cannot be performed simultaneously, then performing either task can be accomplished in any one of m + n ways. Example: Suppose that either a member of the mathematics faculty or a student who is a mathematics major is chosen as a representative to a university committee. How many different choices are there for this representative if there are 37 members of the mathematics faculty and 83 mathematics majors. Solution: There are 37 ways to choose a faculty member, and there are 83 ways to choose a student. Choosing a faculty member is never the same as choosing a student. By the rule of sum there are 37 + 83 = 120 possible choices.

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Discrete Mathematics - Permutations 23-4 The Rule of Product If a procedure can be broken down into two stages, and if there are m possible outcomes of the first stage and if, for each of these outcomes, there are n possible outcomes for the second stage, then the total procedure can be carried out, in the designated order, in m n ways. Example:
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23 - Introduction Permutations Discrete Mathematics Andrei...

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