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Probability and Combinatorics Quiz: Solutions Using Factorials and Powers

This quiz challenges your understanding of combinatorial principles and probability, focusing on the use of factorials and powers. Questions include calculating the minimum number of car purchases needed to guarantee a sale of 5 cars of the same brand, finding coin flip sequences with a specific count of heads, and determining the ways to assign projects to students. Additionally, it explores the concept of set cardinality through powersets. Test your skills and enhance your mathematical reasoning in this engaging session on April 28.

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Probability and Combinatorics Quiz: Solutions Using Factorials and Powers

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  1. Quiz 3 Counting: 4.3, 4.4

  2. 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?

  3. 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.

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