hmwk_7 (1)

# hmwk_7 (1) - (b Give a brief discussion that will convince...

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Discrete Structures Oct 17 2011 Assignment 7: Due at beginning of class Monday, Oct 17th Prof. Hopcroft *******Note: this homework is subject to point evaluations for clarity of writing and clarity of mathematical statements.******** *******Please print out and staple the grade sheet to the back of your homework!****** 1. (a) Use the principle of inclusion and exclusion to determine the number of integers between 1 and 1000, including 1 and 1000, not divisible by 2, 3, 5, or 7. (b) If there are 100 students in a course where the possible grades are A, B, and C, what is the minimum number of students who must have gotten the same grade? What principle did you use to get your answer? 2. Prove that m + n k = k X i =0 m k - i n i 3. (a) What is the set of physical symmetries of the cube? Ie, what are the ways you can rotate, reflect, and/or flip the cube without cutting and pasting it back together?
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Unformatted text preview: (b) Give a brief discussion that will convince us that your answer to part (a) is correct. 4. (a) What are the symmetries of the tic tac toe board game? Ie, what are the ways you can rotate, reﬂect, and/or ﬂip the tic tac toe board, such that the next best move to a board(after it was rotated, reﬂected, etc) is still the next best move after the board was rotated/reﬂected/ﬂipped? (b) Construct the group multiplication table for the symmetries. 5. Consider tic tac toe boards that are completely ﬁlled in with 0’s and × ’s. Use Bernside’s theorem to determine the number of equivalence classes. Note: You may use that a ﬁlled tic tac toe has 5-X’s and 4-O’s or not restrict the number of X’s and O’s. Be sure to specify which assumption you use. 1...
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