hw2_sol.pdf - COMP2121B Discrete Mathematics Homework 2 Due Date Nov 1 2019 Rules Discussion of the problems is permitted but writing the assignment

hw2_sol.pdf - COMP2121B Discrete Mathematics Homework 2 Due...

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COMP2121B: Discrete MathematicsHomework 2Due Date:Nov 1, 2019Rules: Discussion of the problems is permitted, but writing the assignment together is not(i.e. you are not allowed to see the actual pages of another student).Course Outcomes[O1. Abstract Concepts][O2. Proof Techniques][O3. Basic Analysis Techniques]1. (15 points)[O1]Each student at a certain university is given an-digit code, e.g.035...792 (ndigits in total).(a) How many different codes are there?(b) How many codes read the same forward and backward?(Consider the caseswherenis even or odd.)(c) How many codes contain only odd digits?(d) How many codes contain at least one even digit?(e) Consider the case wheren= 6. How many codes have distinct digits? (That is,no digit appears more than once.)Solution:(3 points each)(a) 10n.(b) For evenn, 10n2. For oddn, 10n+12.(c) 5n.(d) From (1c), there are 5ncodes containing no even digit.Thus, 10n-5n(or5n(2n-1)).(e) 10×9×8×7×6×5 = 151200.2. (15 points)[O1]Four sophomores, five juniors and six seniors have volunteered toserve on a committee consisting of five people. In how many ways can the membershipof the committee be selected under the following restrictions respectively?(a) Any of the volunteers can serve on the committee.(b) Only seniors can serve on the committee.(c) At least one senior must serve on the committee.(d) At least two juniors and at least two seniors must serve on the committee.(e) At least one person from each class must serve on the committee, but no morethan two members of the same class can serve on the committee.
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