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Unformatted text preview: Chapter 1 Counting In order to count, there are a few basic strategies that you may want to employ. We will continuously try to point out what you might want to think as you solve these problems. 1.1 Basic Strategies The most important strategy is to break a complicated task into subtasks or cases that are easier to deal with. This may be recursive: subtasks or cases become themselves a task that are then further broken down. 1.1.1 A task can be broken own into a sequence of subtasks If a task can be broken down into a sequence of independent subtasks, one can employ the product rule : Product rule: Suppose that a task can be broken down into a sequence of two independent subtasks where that are n 1 ways to do the first subtask and n 2 ways to do the second, then there are n 1 n 2 ways to do the task. Independent means that the number of choices for the second subtask does not depend on a specific choice for the first subtask. We will point out an example of when subtasks are not independent, later. This naturally extends to the situation where a task can be broken down into a sequence of k subtasks. If you can phrase the task as first do this subtask and then do this subtask and , this indicates a sequence of subtasks. 1.1.2 A task can be split up into cases If a task can be split up into cases, and these cases are exclusive, one can employ the sum rule : Sum rule: If a task can be split up into two cases that are exclusive (non-overlapping) where the first can be done in in n 1 ways and the second in n 2 ways, then there are n 1 + n 2 ways to do the task. This naturally extends to the situation where a task can be split up into k cases that are exclusive. If you can phrase the task as this happens or this happens or , you want to consider cases and can possibly employ the sum rule. 1 2 Chapter 1. Counting 1.1.3 Counting the complement Finally, sometimes it is easier to count the choices that are not the ones you want to count. To then determine the choices you do want to count you answer the following questions: How many choices are there in the universe of discourse? (If you forgot what this is, we will point it out in the context of examples, later.) How many of these choices are not among the events you want to count? By subtracting the second from the first, you are left with the desired count. Rule of complements : Count all possible choices. Count all choices that are NOT in the set that you are trying to count. Subtract. 1.2 Building Your Intuition We will now build your intuition on when to employ what strategy through a series of examples. Example 1 How many different choices of outfits can result if one has two pairs of jeans (blue and red) and three shirts (blue, plaid, and green)....
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This note was uploaded on 03/19/2008 for the course CS 336 taught by Professor Myers during the Spring '08 term at University of Texas at Austin.
- Spring '08