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AMS 301.3 (Fall, 2009)
Estie Arkin
Exam 3 – Solution sketch
Mean 77.9, median 80, high 100 (4 of them!), low 13.
1. (20 points) Build a generating function for
a
r
in the following procedures. Remember to state
which coe±cient solves the initial problem. You do
not
need to calculate the coe±cient.
(a). Select
r
balls from an unlimited pile of white balls, and a pile of 10 red balls, such that the
number of red balls must be even.
Coef of
x
r
in
g
(
x
) = (1 +
x
2
+
x
4
+
...
+
x
10
)(1 +
x
+
x
2
+
....
).
(b). Select 5 letters from a,b,c,d, so that b,c,d each appears at most once and if a appears then it
appears exactly 4 times. (The order in which letters are selected is not important.)
Coef of
x
5
in
g
(
x
) = (1+
x
4
)(1+
x
)
3
. Common mistake: saying that a must appear exactly 4 times.
Not true, it can also not appear at all.
(c). A homework assignment for a calculous class is generated by the computer from a very large list
of problems. The homework set will have 11 problems (the order of the problems is not important),
at least 3 of the problems should be easy problems, at least 3 medium di±culty problems and at
least 2 hard problems. (Note: do not distinguish between problems of the same di±culty level, i.e.,
you only care how many problems of each di±culty level are on the homework assignment.)
Coef of
x
11
in
g
(
x
) = (
x
3
+
x
4
+
...
)
2
(
x
2
+
x
3
+
....
).
(d). A survey team wants to select 6 students to interview from 4 possible universities. There
should be at most 3 students from Binghamton, at most 2 students from Albany, at most 2 from
Bu²alo and any number of students from Stony Brook. (Note: Assume that for the purposes of
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 Spring '11
 Estie

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