Discrete Mathematics and Its Applications with MathZone

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UCI ICS/Math 6A, Summer 2007 6-Counting-1 [email protected], after [email protected] Counting: Product and Sum Rules Product Rule: Assuming a) We need to perform procedure 1 AND procedure 2. b) There are n 1 ways to perform procedure 1 and c) n 2 ways to perform procedure 2. There are n 1 n 2 ways to perform procedure 1 AND procedure 2. Sum Rule: Assuming a) We need to perform procedure 1 OR procedure 2. b) There are n 1 ways to perform procedure 1 and c) n 2 ways to perform procedure 2. There are n 1 + n 2 ways to perform procedure 1 OR procedure 2. This “OR” is an “exclusive OR.” One choice or the other, but not both.
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UCI ICS/Math 6A, Summer 2007 6-Counting-2 [email protected], after [email protected] Counting Examples: Product Rule Q. How many ways can we label a chair if each label consists of both a letter AND a number between 1 and 50, inclusive? There are 26 possibilities for the letter AND 50 for the number… A: 26x50=1300 Q. With 31 flavors of ice cream, 4 sizes of serving, and a choice of “cone” or “dish,” how many different orders of ice cream are there? A: 31x4x2=248 Q. What is the total number of subsets of a set A with n elements? That is, |P(A)| = ____? A: Representing each subset of A as a bitstring of length n (A=1111111…1, empty set=00000...0), |P(A)|=2 n .
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UCI ICS/Math 6A, Summer 2007 6-Counting-3 [email protected], after [email protected] Counting Examples: Sum Rule Q: There is one position available for a faculty job at Irvine. The applicant must come from either Berkeley which has 20 candidates or UCLA which has 50 candidates. What is the total number of possible candidates for the position? A: The applicant must be from Berkeley OR UCLA. So we have 20+50=70 possible applicants. Q: At a resaurant, there are 18 dinners with meat, 10 different dinners with fish, and 5 vegetarian dinners?
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