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# Counting - Definesamplingwith//unordered kpermutations...

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Learning Objectives Define sampling with / without replacement and order / unordered  and relate to counting methods  Calculate probabilities using the counting principle by selecting the  correct method or combination of methods: k-permutations combinations partitions

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Example Polymers are to be created from three kinds of  subunits.  There are 4 subunits of type A, 3 of type  B, and 2 of type C.  How many different 9-unit  polymers are possible?
Counting Principle Consider a process that consists of r stages.  Suppose that: There are n 1  possible results at the first stage For every possible result of the first stage, there are n 2  possible results at the  second stage More generally, for any possible results of the first i-1 stages, there are n i   possible results at the ith stage Then, the total number of possible results of the r-stage process is: n 1 n 2 …n r Example: roll 2 dice – 6*6=36 possible results

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Exercises Exercise #18 Exercise #19
Selection of k out of n objects Do we care about the order in which objects are selected? Ordered UnOrdered example: order of the cards you draw in poker doesn’t matter Do we get to reuse objects? With Replacement (reuse allowed) Without Replacement (no reuse)

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Approaches Ordered UnOrdered With Replacement (reuse) Without Replacement (no reuse) Permutations Combinations n ! ( n - k )! n * n *... = n k n k ae è ç ö ø ÷ = n !
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Counting - Definesamplingwith//unordered kpermutations...

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