Quiz 3 Counting: 4.3, 4.4

Quiz3: April 28, 3.30-3.45 pm Your answers should be expressed in terms of factorials and/or powers. 1) A car dealer has 20 Chevy’s, 20 BMWs, 20 Pontiacs and 20 Honda’s. How many customers must buy a car on a particular day to be sure that the dealer sells at least 5 cars of the same brand (on that day)? 2) Imagine we flip a coin 10 times. How sequences are there that have 1 or 2 heads (and the rest tails)? 3) There are 10 students in class. Each student is required to do a different project, and there are 20 projects to choose between. In how many ways can we assign projects to students? 4) Explain why the following equation proofs that the cardinality of a set with n elements is 2^n?

Quiz3: April 28, Answers 1) A car dealer has 20 Chevy’s, 20 BMWs, 20 Pontiacs and 20 Honda’s. How many customers must buy a car on a particular day to be sure that the dealer sells at least 5 cars of the same brand (on that day)? Pigeonhole principle: 17 cars (4 brands, each 4 cars makes 16 cars, add one more) 2) Imagine we flip a coin 10 times. How sequences are there that have 1 or 2 heads (and the rest tails)? 1 head: C(10,1), 2 heads: C(10,2). Sum-rule: C(10,1)+C(10,2). 3) There are 10 students in class. Each student is required to do a different project, and there are 20 projects to choose between. In how many ways can we assign projects to students? First student has 20 choices, second 19 etc. Total P(20,10)=20!/10! 4) Explain why the following equation proofs that the cardinality of a set with n elements is 2^n? C(n,k) is the number of ways to choose a subset of k elements out of n elements. Powerset counts all subsets so by the sum-rule we need to add them.