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mod6 - Module 6 Basic Counting Theme 1 Basic Counting...

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Module 6: Basic Counting Theme 1: Basic Counting Principle We start with two basic counting principles, namely, the sum rule and the multiplication rule . The Sum Rule : If there are different objects in the first set , objects in the second set , , objects in the th set , and if the sets are disjoint (i.e., for any ), then the total number of ways to select an object from one of the set is in other words, The Multiplication Rule : Suppose a procedure can be broken into successive (or- dered) stages, with outcomes in the first stage, outcomes in the second stage, , outcomes in the th stage. If the number of outcomes at each stage is independent of the choices in previous stages, and if the composite outcomes are all distinct, then the total procedure has different composite outcomes. Sometimes this rule can be phrased in terms of sets as follows Example 1 : There are students in an algebra class and students in a geometry class. How many different students are in both classes combined? This problem is not well formulated and cannot be answered unless we are told how many students are taking both algebra and geometry. If there is not student taking both algebra and geometry, then by the sum rule the answer is . But let us assume that there are students taking both algebra and geometry. Then there are students only in algebra, students only in geometry, and students in both algebra and geometry. Therefore, by the sum rule the total number of students is . Example 2 : There are boxes in a postal office labeled with an English letter (out of English char- acters) and a positive integer not exceeding . How many boxes with different labels are possible? 1
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The procedure of labeling boxes consists of two successive stages. In the first stage we assign different English letters, and in the the second stage we assign natural numbers (the second stage does not depend on the outcome of the first stage). Thus by the multiplication rule we have different labels. Example 3 : How many different bit strings are there of length five? We have here a procedure that assigns two values (i.e., zero or one) in five stages. Therefore, by the multiplication rule we have different strings. Exercise 6A : How many binary strings of length are there that start with a and end with a ? Example 4 : Counting Functions . Let us consider functions from a set with elements to a set with elements. How many such functions are there? We can view this as a procedure of successive stages with outcomes in each stage, where the outcome of the next stage does not depend on the outcomes of the previous stages. By the multiplication rule there are functions. But, let us now count the number of one-to-one functions from a set of elements to the set of elements. Again, we deal here with a procedure of successive stages. In the first stage we can assign values. But in the second stage we can only assign values since for a one-to- one function we are not allowed to select the value used before. In general, in the th stage we have only elements at our disposal. Thus by (a generalized) multiplication rule we have
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