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# lecture-note - 1 COMBINATORIAL ANALYSIS 1.1 Counting...

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1. COMBINATORIAL ANALYSIS 1.1 Counting Principles 1. Theorem (The basic principle of counting): If the set E contains n elements and the set F contains m elements, there are nm ways in which we can choose, first, an element of E and then an element of F . 2. Theorem (The generalized basic principle of counting): If r experiments that are to be performed are such that the first one may result in any of 1 n possible outcomes, and if for each of these 1 n possible outcomes there are 2 n possible outcomes of the second experiment, and if for each of the possible outcomes of the first two experiments there are 3 n possible outcomes of the third experiment, and if …, then there is a total of r n n n 2 1 , possible outcomes of the r experiments. 3. Theorem: A set with n elements has n 2 subsets. 4. Tree diagrams 1.2 Permutations 1. Permutation: ! n 1

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The number of permutations of n things taken r at a time: )! ( ! r n n P n r 2. Theorem: The number of distinguishable permutations of n objects of k different types, where 1 n are alike, 2 n are alike, …, k n are alike and k n n n n 2 1 , is ! ! ! ! 2 1 k n n n n 1.3 Combinations 1. Combination: The number of combinations of n things taken r at a time: )! ( ! ! r n r n C n r (combinatorial coefficient; binomial coefficient) 2. Binomial theorem: n i i n i n i n y x C y x 0 ) ( 3. Multinomial expansion: In the expansion of n k x x x ) ( 2 1 , the coefficient of the term k n k n n x x x 2 1 2 1 , n n n n k 2 1 , is ! ! ! ! 2 1 k n n n n . Therefore, n n n n n k n n k n k k k x x x n n n n x x x 2 1 2 1 2 1 2 1 2 1 ! ! ! ! ) ( . Note that the sum is taken over all nonnegative integers 1 n , 2 n , …, k n such that n n n n k 2 1 . 1.4 The Number of Integer Solutions of Equations 1. There are 1 1 n r C distinct positive integer-valued vectors ) , , , ( 2 1 r x x x satisfying n x x x r 2 1 , 0 i x , r i , , 2 , 1 . 2. There are 1 1 r n r C distinct nonnegative integer-valued vectors ) , , , ( 2 1 r x x x satisfying n x x x r 2 1 . 2
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2. AXIOMS OF PROBABILITY 2.1 Sample Space and Events 1. Set theory concepts: set, element, roster method, rule method, subset, null set (empty set). 2. Complement: The complement of an event A with respect to S is the subset of all elements of S that are not in A . We denote the complement of A by the symbol A’ ( c A ). 3. Intersection: The intersection of two events A and B , denoted by the symbol B A , is the event containing all elements that are common to A and B . -- Two events A and B are mutually exclusive, or disjoint, if B A that is, if A and B have no elements in common.
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