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Unformatted text preview: 2. Determine the generating function for the number of ways
to distribute 35 pennies {from an unlimited supply} among ﬁve
children if {a} there are no restrictions; (h) each child gets at
least 11:; {c} each child gets at least 21:; {d} the oldest child gets
at least lite; and (e) the two youngest children must each get at
least 101:. 4. 3) Explain why the generating function for the number of
ways to have :1 cents in pennies and nickels is {1+x+x2+x3+}(l+x5+x'"+}. 2. Detenuine the sequence generated by each of the following
generating functions. a) f(I}= {Ex—3F b} for} =x‘ftl —x)
c} for) =x3ﬂil erg} d) for} = 1m +3x}
e) fix} = lft3x) r) fix1=l;(1—x)+3x"11 II. Determine the constant (that is, the coefﬁcient of x”} in
{3x2  (2a))”. 15. In how many ways can Traci select :2 marbles from a large
supply of blue. red, and yellow marbles {all of the same size] if
the selection must include an even number of blue ones? 6. What is the generating function for the number of partitions
of n E N into summands that {a} cannot occur more than ﬁve
times; and {b} cannot exceed 12 and cannot occur more than
ﬁve times? 8. Show that the number of partitions of n IE 2+ where no
summand is divisible by 4 equals the number of partitions of n
where no even summarid is repeated {although odd stunmands
may' or mayr not be repeated). ...
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 Spring '08
 PEARCE

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