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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 checkerboardcovering is called
magical
if it contains exactly one octarine
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This note was uploaded on 01/12/2012 for the course MATH 681 taught by Professor Wildstrom during the Fall '09 term at University of Louisville.
 Fall '09
 WILDSTROM
 Binomial

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