By Alan Doerr, Kenneth Levasseur

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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.

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