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MIT6_042JF10_final

MIT6_042JF10_final - 6.042/18.062J Mathematics for Computer...

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6.042/18.062J Mathematics for Computer Science December 14, 2010 Tom Leighton and Marten van Dijk Final Name: This quiz is closed book , but you may have two 8.5” × 11” sheets with notes in your own handwriting on both sides. Calculators are not allowed. You may assume all of the results presented in class. Please show your work. Partial credit cannot be given for a wrong answer if your work isn’t shown. Write your solutions in the space provided. If you need more space, write on the back of the sheet containing the problem. Please keep your entire answer to a prob- lem on that problem’s page. Be neat and write legibly. You will be graded not only on the correctness of your answers, but also on the clarity with which you express them. If you get stuck on a problem, move on to others. The problems are not arranged in order of difficulty. The exam ends at 4:30 PM. Problem Points Grade Grader Problem Points Grade Grader 1 20 8 20 2 12 9 11 3 10 10 10 4 10 11 10 5 15 12 32 6 10 13 10 7 10 total 180
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2 Final Problem 1. [20 points] We define the following recurrence for n 0: T n + 2 = T n + 1 + 2 T n where T 0 = T 1 = 1. (a) [8 pts] Prove by induction that T n is odd for n 0. You do not need to solve the recurrence for this.
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3 Final (b) [12 pts] Prove by induction that gcd ( T n + 1 , T n ) = 1 for n 0. You may assume that T n is odd for all n . You do not need to solve the recurrence for this.
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4 Final Problem 2. [12 points] Find a closed-form solution to the following recurrence: x 0 = 4 x 1 = 23 x n = 11 x n 1 30 x n 2 for n 2.
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5 Final Problem 3. [10 points] Note: in this question, you may use “choose” notation or factorials in your answers for both (a) and (b). In the card game of bridge, you are dealt a hand of 13 cards from the standard 52-card deck. (a) [5 pts] A balanced hand is one in which a player has roughly the same number of cards in each suit. How many different hands are there where the player has 4 cards in one suit and 3 cards in each of the other suits? (b) [5 pts] Not surprisingly, a non-balanced hand is one in which a player has more cards in some suits than others. Hands that are very desired are ones where over half the cards are in one suit. How many different hands are there where there are exactly 7 cards in one suit?
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6 Final Problem 4. [10 points]
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