AMS 301.3 (Fall, 2007)
Estie Arkin
Exam 3 – Solution sketch
Mean 82.32, median 85, high 100 (2 of them!), low 43.
1. (20 points) Build a generating function for the the number of election outcomes in the race for
class president if there are 4 candidates and 30 students votes. (You do
not
need to calculate the
coefficient.): Each part is independent of the others!
(a).
Every candidate gets at least one vote?
Let
e
i
be the number of votes for candidate
i
, so
e
1
+
e
2
+
e
3
+
e
4
= 30,
e
i
≥
1, gives the generating function
g
(
x
) = (
x
+
x
2
+
x
3
+
· · ·
)
4
, we want
the coefficient of
x
30
.
(b). Tiffany (one of the candidates) gets at least 4 votes?
g
(
x
) = (1+
x
+
x
2
+
x
3
+
· · ·
)
3
(
x
4
+
x
5
+
· · ·
),
we want the coefficient of
x
30
.
(c). Voters are allowed to abstain (vote for none of the 4 candidates)? Let
e
5
be the number of
voters that abstain, we have
e
1
+
e
2
+
e
3
+
e
4
+
e
5
= 30,
e
i
≥
0 and so the generating function is
g
(
x
) = (1 +
x
+
x
2
+
x
3
+
· · ·
)
5
, we want the coefficient of
x
30
.
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 Spring '08
 ARKIN
 Recurrence relation, Generating function, Estie Arkin, Extra cheese, a1 b5

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