Fundamental-Principles-of-Counting-1.pptm

Fundamental-Principles-of-Counting-1.pptm - Chapter I...

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FUNDAMENTAL PRINCIPLES OF COUNTING Chapter I
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INTRODUCTION Enumeration or counting is a basic mathematical tool that has been used in most diverse circumstances. The applications range from basic daily life activities to coding theory, probability and statistics, and in the analysis of most challenging algorithms. There are few terminologies, concepts, rules and laws should be understood prior going deep in Discrete Structures (DS) topics. In DS sense, counting is the rule for finding number of possible outcomes.
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THE RULES OF SUM AND PRODUCT 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 can not be performed simultaneously, then performing either task can be accomplished in any one of m + n ways. Simply speaking: what can be done is task (procedure) 1 OR task (procedure) 2. This can be generalized to multiple tasks.
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THE RULE OF SUM Example 1.1 : At a restaurant, there are 18 dinner sets with meat and 10 sets with fish. How many dinner sets to choose from? Solution : Each set can be meat (m) OR fish (n), thus, we can have m + n ways or 18 + 10 = 28 choices.
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THE RULE OF SUM (CONT…) Example 1.2 : A student has the right to borrow three science books, two novels and four literature books from the library. How many choices does the student have at any time? Answer: We have three options, science books (m), novels (n), literature books (o); thus, The number of ways or choices can be made at a time is m + n + o = 3 + 2 + 4 = 9.
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THE RULES OF PRODUCT If a procedure can be broken down into first and second stages, and if there are m possible outcomes for 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 our, in the designated order, in mn ways.
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  • Winter '16
  • Andrzej RUSZEWSKI
  • Combinatorics, Grammatical person

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