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MIT6_042JF10_final

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

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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.
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.
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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