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Question: discrete structures questions 1 and 2...

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Discrete structures
Questions 1 and 2
1. A restaurant has 3 appetizers, 7 main dishes, and 2 desserts. a) How many ways can you order 1 item? b) How many ways can you order two items from different categories? c) How many ways can you order three items from different categories? d) How many ways can you order up to three items from different categories? 2. How many integers between 1 and 5000 (inclusive) are not divisible by any of 2, 3, or 5? 3. Let Q be the set of 7-digit strings (order matters) using digits 0-9 with no repeated digits. a) What is the cardinality of Q? b) How many strings in Q contain a 3? c) How many strings in Q contain a 3 and a 5? d) How many strings in Q contain a 4 or a 6, but not both? Two students are working on a problem that asks them to count 3-digit strings using 0-9, with repetition allowed, where at least one of the digits is a 1. Unfortunately, each student has an error in their reasoning. The first student says, My answer is 300. I used the Additive Principle. I let set A to consist of those strings where the first digit is 1 such as 123 or 145. Set B consists of those with the second digit equal to 1, and set C consists of those with the third digit equal to 1. Since at least one digit has to equal 1, AUBUC gives me the answer. To find the number of elements in A, I found that there are 10 possible options for each of the second and last 4. a) digits, so there are a total of 100 elements in A. Similarly 100 elements in B and 100 elements C. Therefore, by Additive Principle
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