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exam-2-091112

# exam-2-091112 - MATH 681 Exam#2 Answer exactly four of the...

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MATH 681 Exam #2 Answer exactly four of the following six questions. Indicate which four you would like graded! Binomial coefficients, Stirling numbers, and arithmetic expressions need not be simplified in your answers. 1. (10 points) A necklace consists of 6 gems; the gems can be garnets, tourmaline, or zircons. Two necklaces are considered to be identical if one can be obtained by rotating or flipping the other. (a) (5 points) Find a pattern inventory for all such necklaces. You need not alge- braically expand the pattern inventory. (b) (5 points) Either using your pattern inventory or by other means, determine the number of necklaces which use two of each gemstone. 2. (10 points) A 1 × n checkerboard is to be covered with dominoes (which cover two squares) and checkers (which cover one each). We have dominoes in four colors: green, yellow, purple, and octarine, and checkers in four other colors: black, white, cyan, and red. A checkerboard-covering is called magical if it contains exactly one octarine domino, and to the right of the octarine domino uses only checkers.

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