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Unformatted text preview: Combinations with repetition Periklis A. Papakonstantinou ? University of Toronto We discuss how to count the number of distinct combinations with repetition of r objects from n distinct objects (this is also discussed in section 6.5 in the text). On the way we make explicit reference to the technique of reduction for counting problems. Counting problems are one of the few places in Discrete Mathematics where we allow arguments to be somehow “hand-waving”. As in every area of mathematics counting can be also formalized through formal power series, finite group theory etc. At this level we consider this to be an unnecessary complication. Nevertheless, as simple as counting seems to be it is a creative and challenging task even for problems that at first glance appear to be very simple. How do we count? You have seen two general axioms. The rule of product and the rule of sum. Just applying these two we can solve every counting problem. But without knowing anything else this task can become from complicated to very complicated. Therefore, we derive (from these two simple rules) some others which we use when counting as a black box. For example, you already know formulas counting the number of distinct arrangements of distinct objects, a formula that counts the number of distinct permutations, a formula for combinations and so on. Actually, even at the end of this course you won’t know more than 6-7 such formulas. And it is true that only very few counting problems can be solved by a direct application of such a formula. So how do we count? Here is a general receipt: 1. We break the counting problem into subproblems which altogether are connected to the main problem through the rule of product or sum. 2. We solve each subproblem. 3. Then, we use the rule of product or sum to derive the result we were looking for. How do we solve each subproblem? Either this subproblem is something that we know how to solve or we recursively apply the above procedure for this subproblem. In this handout we do not encounter a complicated counting problem that we break it into many subproblems in this way. Our main concern is the phrase: “that we know how to solve” . Each subproblem can be tackled from elementary principles or by using a technique which reduces the counting (sub)problem that we do not know how to solve to a counting (sub)problem that we know how to solve. Such reductions are the main topic of this document. ? This document possibly contains typos. Please submit any typos, remarks, suggestions to [email protected] . 2 Periklis Papakonstantinou (ECE 190, Fall 2006) Size-preserving reductions/transformations Restricted classes of functions We recall that a function f : A → B , with domain A and range B is a relation that maps each element of the domain to one element of the range. Here we introduce three restrictions to the class of functions from A to B . The last one is of main interest to us.....
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- Fall '06
- Finite set, Bijection, distinct objects, bijective transformation