CS2800-Counting_v.4 - 1 Discrete Math CS 280 Prof. Bart...

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Unformatted text preview: 1 Discrete Math CS 280 Prof. Bart Selman selman@cs.cornell.edu Module Counting Combinatorics Count the number of ways to put things together into various combinations. e.g. If a password is 6, 7, or 8 characters long; a character is an uppercase letters or a digit, and the password is required to include at least one digit - how many passwords can there be? Or, how many graphs are there on N nodes? How many of those are 3-colorable? Many uses in discrete math (because of all the discrete strucures), including e.g. probability theory (next topic). E.g., what is the probability that a randomly generated graph is 3- colorable? First, two most basic rules: Sum rule Product rule How can we figure that out? Sum Rule Let us consider two tasks: m is the number of ways to do task 1 n is the number of ways to do task 2 Tasks are independent of each other, i.e., Performing task 1 does not accomplish task 2 and vice versa. Sum rule : the number of ways that either task 1 or task 2 can be done, but not both , is m + n . Generalizes to multiple tasks ... Task 1 Task 2 4 Example A student can choose a computer project from one of three lists. The three lists contain 23, 15, and 19 possible projects respectively. How many possible projects are there to choose from? 23+15+19 Ok not to worry. things will get more exciting... 5 Sum rule example How many strings of 4 decimal digits, have exactly three digits that are 9s? The string can have: The non-9 as the first digit OR the non-9 as the second digit OR the non-9 as the third digit OR the non-9 as the fourth digit Thus, we use the sum rule For each of those cases, there are 9 possibilities for the non-9 digit (any number other than 9) Thus, the answer is 9+9+9+9 = 36 6 Set Theoretic Version If A is the set of ways to do task 1, and B the set of ways to do task 2, and if A and B are disjoint , then: the ways to do either task 1 or 2 are A B , and | A B | = | A | + | B | Product Rule Let us consider two tasks: m is the number of ways to do task 1 n is the number of ways to do task 2 Tasks are independent of each other, i.e., Performing task 1does not accomplish task 2 and vice versa. Product rule : the number of ways that both tasks 1 and 2 can be done in mn . Generalizes to multiple tasks ... task 1 task 2 8 Product rule example There are 18 math majors and 325 CS majors How many ways are there to pick one math major and one CS major? Total is 18 * 325 = 5850 Product Rule How many functions are there from set A to set B? A B To define each function we have to make 3 choices, one for each element of A. Each has 4 options (to select an element from B)....
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CS2800-Counting_v.4 - 1 Discrete Math CS 280 Prof. Bart...

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