Combinatorial_Analysis

Combinatorial_Analysis - 2.2 Combinatorial analysis...

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2.2 Combinatorial analysis Department of Electrical and Computer Engineering McGill University, Montr´eal, CANADA September 2006 B. Champagne Chapter 1 – p. 1/2 5
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What is combinatorial analysis? Part of mathematics dealing with the study and development of systematic methods for counting. Find applications in many areas of sciences and engineering: probability and statistics, information theory, data compression, genetics, etc. The calculation of probabilities often amounts to counting the number of elements in various sets. Combinatorial techniques will be of great help in the solution of these problems. B. Champagne Chapter 1 – p. 2/2 5
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2.2.1 Basic counting techniques B. Champagne Chapter 1 – p. 3/2 5
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Generalized counting principle Definition: A r -tuple is an ordered list (or vector) of elements, of the form ( x 1 , x 2 , ..., x r ) , or simply x 1 x 2 ...x r Two r -tuples are equal ( = ) if and only if each of the corresponding elements are identical. Theorem 2.4 : Let A be a set of r -tuples, { x 1 x 2 ... x r } , such that there are: Firstly: n 1 different ways in which to chose x 1 , Secondly: n 2 different ways in which to chose x 2 , ... Finally: n r different ways in which to chose x r . Then A contains N ( A ) = n 1 n 2 ... n r different r -tuples. B. Champagne Chapter 1 – p. 4/2 5
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Note: The theorem specifies only the number of possible choices available at each step the specific choices in the r th step may depend on previous choices, but not their number n r . Corollary: Suppose the sets A 1 , A 2 , ..., A r contain n 1 , n 2 , ..., n r elements, respectively. Then the product set A 1 × A 2 × ... × A r = { ( a 1 , a 2 , ..., a r ) : a i A i } (1) contains n 1 n 2 ...n r elements. B. Champagne Chapter 1 – p. 5/2 5
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Example In Quebec, license plate numbers are made up of 3 letters followed by 3 digits, that is l 1 l 2 l 3 d 1 d 2 d 3 where l i is any one of 26 possible letters from a to z, and d i is any one of the possible digits from 0 to 9. Thus there are, in principle, 26 × 26 × 26 × 10 × 10 × 10 = 26 3 × 10 3 = 17 , 576 , 000 (2) different license plate numbers.
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Combinatorial_Analysis - 2.2 Combinatorial analysis...

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