Com S 280 - Counting

# Com S 280 - Counting - o the sum rule a basic counting...

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Counting - Definitions o combinatorics: the study of arrangement of objects o enumeration: the counting of arrangements of objects o tree diagram: a diagram made up of a root, branches leaving the root, and other branches leaving some of the endpoints of branches o permutation: an ordered arrangement of the elements of a set o r-permutation: an ordered arrangement of r elements of a set o ) , ( r n P : the number of r-permutations of a set with n elements o r-combination: an unordered selection of r elements of a set o ) , ( r n C : the number of r-combinations of a set with n elements o r n (binomial coefficient): the number of r-combinations of a set with n elements - Results o the product rule: a basic counting technique, which states that the number of ways to do a procedure that consists of two subtasks is the product of the number of ways to do the first task and the number of ways to do the second task after the first task has been done
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Unformatted text preview: o the sum rule: a basic counting technique, which states that the number of ways to do a task in one of two ways is the sum of the number of ways to do these tasks if they cannot be done simultaneously o the pigeonhole principle: When more than k objects are placed in k boxes, there must be a box containing more than one object o the generalized pigeonhole principle: When N objects are placed in k boxes, there must be a box containing at least k N / objects o ( 29 ( 29 ! ! , r n n r n P-= o ( 29 ( 29 ! ! ! , r n r n r n r n C-= = o Pascal’s Identity: + -= + k n k n k n 1 1 o the Binomial Theorem: ( 29 ∑ =- = + n k k k n n y x k n y x...
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