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L04_F10 - AMS 311 Lecture 4 Spring 2010 Generalized Basic...

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AMS 311: Lecture 4 Spring 2010 Generalized Basic Principle of Counting : If r actions that are to be performed are such that the first one may result in 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. 1.4. Combinations There are r n C combinations of r elements chosen from n objects, where ! )! ( ! r r n n C r n - = The binomial theorem: For any integer 0 n , ( 29 = - = + n k k n k n k n y x y x 0 ) ( 1.5 Multinomial Coefficients If , 2 1 n n n n r = + + + we define ( 29 . ! ! ! ! 2 1 , , , 2 1 r n n n n n n n n r = It represents the number of possible divisions of n distinct objects into r distinct groups of respective sizes . , , , 2 1 r n n n It is a multinomial coefficient.
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