# Berkeley MATH 55 - Lecture 14 (8 pages)

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## Lecture 14

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- Pages:
- 8
- School:
- University of California, Berkeley
- Course:
- Math 55 - Discrete Mathematics

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Math 55 Spring 2004 Lecture notes 14 March 9 Tuesday Read Chapter 4 Goals for Today Continue counting principles Tree diagrams Pigeonhole principle Permutations Combinations Homework 1 Based on a recent 1st grade homework assignment from a local school Perhaps this is covered later 1 1 Suppose you can have 9 pieces of fruit which can be either apples or oranges How many ways can you have n pieces of fruit for example you could have 4 apples and 5 oranges or 0 apples and 9 oranges etc 1 2 Answer 1 1 for n pieces of fruit 1 3 Suppose you can have 9 pieces of fruit which can be apples oranges or pears How many ways can you have 9 pieces of fruit actual 1st grade assignment 1 4 Answer 1 3 for n pieces of fruit 1 5 Suppose you can have n pieces of fruit and there are m different kinds of fruit How many ways can you have n pieces of fruit 2 Also based on a recent 1st grade homework assignment Suppose you can make shapes from putting together identical equilateral triangles edge to edge Triangles must not overlap and if they are adjacent they must line up along an entire edge In the picture below the leftmost shape is ok and the other two are not Shapes must be connected i e you can get from any triangle to any other triangle by moving across adjacent edges How many different shapes using 2 3 4 5 6 and 7 triangles are there Shapes are considered the same if one shape can be slid or rotated but not turned over to lie exactly on top of the other shape The 6 triangle problem was the actual 1st grade assignment See if you can do 7 1 3 4 5 6 7 Review 1 The If S1 4 14 24 34 44 5 20 10 18 10 20 42 14 24 38 54 Counting Principles Sum Rule S1 and S2 are disjoint sets then the number of members of U S2 is S1 U S2 S1 S2 2 Inclusion Exclusion Principle If S1 and S2 are arbitrary sets then S1 U S2 S1 S2 S1 inter S2 3 The Product Rule If S1 and S2 are sets and S1 x S2 s1 s2 s1 in S1 and s2 in S2 is the Cartesian product of S1 and S2 then S1 x S2 S1 x S2 If S1 S2 Sk are sets S1 x S2 x Sk

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