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Methods of Enumeration Counting tools can be very important in probability … particularly if you have a finite sample space with equally likely outcomes Counting tools can be very important in probability … particularly if you have a finite sample space with equally likely outcomes Multiplication Principle (or Product Rule) Multiplication Principle (or Product Rule) An experiment consisting of m steps with n i possible outcomes on the i th step has a total of n 1 × n 2 × … × n m possible outcomes An experiment consisting of m steps with n i possible outcomes on the i th step has a total of n 1 × n 2 × … × n m possible outcomes Tree Diagram Tree Diagram

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Sampling Terminology Sampling with Replacement Sampling with Replacement Sampling where an object is selected and replaced before the next object is selected. The probability of selection remains constant. Sampling where an object is selected and replaced before the next object is selected. The probability of selection remains constant. Sampling without Replacement Sampling without Replacement Sampling where an object is not replaced after it is selected. The probability of selection changes after each selection. Sampling where an object is not replaced after it is selected. The probability of selection changes after each selection.

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Order is Important Ordered Sample of Size r Ordered Sample of Size r r objects are selected from n objects, and the order is important r objects are selected from n objects, and the order is important Sampling with Replacement Sampling with Replacement There are n r possible ordered samples when sampling r objects from a set of n objects with replacement. There are n r possible ordered samples when sampling r objects from a set of n objects with replacement.

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Order is Important Sampling without Replacement Sampling without Replacement There are n! different ways in which n objects can be ordered. Each possible ordering is called a permutation. There are n! different ways in which n objects can be ordered. Each possible ordering is called a permutation. Permutation Rule Permutation Rule For rn, the number of permutations of r objects selected from n objects is For rn, the number of permutations of r objects selected from n objects is

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Order is not Important Order of selection is often not important. If so, then all r! orders of r objects selected from n objects are considered the same. Order of selection is often not important. If so, then all r! orders of r objects selected from n objects are considered the same. Sampling without Replacement Sampling without Replacement Combination Rule Combination Rule For rn, the number of combinations of r objects selected from n objects is For rn, the number of combinations of r objects selected from n objects is

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Binomial Coefficient n C r is equivalent to the binomial coefficient, which is the coefficient in the k th expansion of (a+b) n as expressed below: n C r is equivalent to the binomial coefficient, which is the coefficient in the k th expansion of (a+b) n as expressed below: Pascals Triangle Pascals Triangle Pascals Triangle Pascals Triangle

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Distinguishable Permutations Binomial Coefficient Binomial Coefficient Useful for counting the number of distinguishable permutations in a set of two types with r of one type and (n-r) of the other. Useful for counting the number of distinguishable permutations in a set of two types with r of one type and (n-r) of the other. Multinomial Coefficient Multinomial Coefficient Userful for counting the number of distinguishable permutations in a set of s types where there are n i objects of the i th type (n=n 1 +n 2 +…+n s ) Userful for counting the number of distinguishable permutations in a set of s types where there are n i objects of the i th type (n=n 1 +n 2 +…+n s )

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How many ways are there to pass through city A where the arrows represent one-way streets? Answer: mn ways The counting principal: Suppose two experiments.

How many ways are there to pass through city A where the arrows represent one-way streets? Answer: mn ways The counting principal: Suppose two experiments.

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