By Alan Doerr, Kenneth Levasseur
Textbook from UMass Lowell, model 3.0
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Extra resources for Applied Discrete Structures
12. How many integers from 100 to 999 can be written with no 7’s? 13. Consider three persons, A, B, and C, who are to be seated in a row of three chairs. Suppose A and B are identical twins. How many seating arrangements of these persons can there be a If you are a total stranger? b If you are A and B’s mother? This problem is designed to show you that diﬀerent people can have diﬀerent correct answers to the same problem. 14. How many ways can a student do a ten-question true-false exam if he or she can choose not to answer any number of questions?
This is a combination problem, because the order in which the heads appear does not matter. We can think of this as a situation involving sets by considering the set of flips of the coin, 1 through 5, in which heads comes up. The number of ways to get three heads is 5 5·4 3 = 2·1 = 10. 6 (Listing Five Flips, taking order into account). Determine the total number of ways a fair coin can land if tossed five consecutive times. The five tosses can produce any one of the following mutually exclusive, disjoint events: 5 heads, 4 heads, 3 heads, 2 heads, 1 head, or 0 heads.
Do not underestimate the usefulness of simple ideas. 7 (Power Set Cardinality Theorem). If A is a finite set, then |P(A)| = 2|A| . 26 CHAPTER 2. COMBINATORICS Proof. Proof: Consider how we might determine any B 2 P(A), where |A| = n. For each element x 2 A there are two choices, either x 2 B or x 2 / B. Since there are n elements of A we have, by the rule of products, 2 · 2 · · · · · 2 = 2n n factors diﬀerent subsets of A. Therefore, P(A) = 2n . 3 Exercises A Exercises 1. In horse racing, to bet the “daily double” is to select the winners of the first two races of the day.