# final(1)

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Discrete Structures. CSCI-150. Summer 2015. Final Exam. You are allowed to use a formula sheet: It should be a single standard sheet of paper (of Letter/A4 size). The formula sheet has to be handed in after the exam. For every problem, provide a reasonably complete argument supporting your answer. Problem 1 (3 pts) Prove: c ( a b ) ( h → ¬ a ) ( r s ) ( ¬ r ∨ ¬ s ) ( h ∨ ¬ q ) p q ¬ c Problem 2 (3 pts) Prove that for all integers n 3 : 2 2 + 3 2 + 4 2 + 5 2 + . . . + n - 1 2 = n 3 . Problem 3 (1 pts) In how many ways can you buy 20 apples if you can choose between Golden Delicious, Gala, and Fuji? (Assuming that the apples of the same variety are identical). Problem 4 (3 pts) Prove that for all non-negative integers n : 3 | n 3 + 5 n. Problem 5 (2 pts)

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