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MATH 239 Spring 2010 Assignment 3
Due: May 28, 2010, 11:45AM EDT
1. In the Kingdom of Shallitland, King Jeffrey XVI declared that citizens of his coun
try (the Shallites) must use coins of these denominations: 1 byte, 7 bytes, 57 bytes,
80 bytes. Let
n
be a positive integer.
(a)
(2 marks)
Assuming an endless supply of coins available in each denomina
tion, determine the number of ways that a collection of Shallite coins can
make up
n
bytes. Write this as the coefﬁcient in a rational function.
(b)
(2 marks)
King Jeffrey XVI ordered the 7byte coins to be made of expensive
silicon, and made a restriction that each citizen may possess at most 8 of these
7byte coins. Given this restriction, determine the number of ways that one
Shallite citizen can use their own coins to make up
n
bytes. Write this as the
coefﬁcient in a rational function.
(c)
(2 marks)
Let
a
n
and
b
n
be the answers for parts (a) and (b) respectively. Using
your answers in (a) and (b), show that
a
n
≥
b
n
for all
n
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 Spring '09
 M.PEI
 Math

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