AMS 301.2 (Spring, 2010)
Estie Arkin
Exam 3 – Solution sketch
Mean 75.89, median 80, top quartile 87, high 100 (3 of them!), low 18.
1. (20 points) Build a generating function for
a
r
in the following procedures. Remember to state
which coeFcient solves the initial problem. You do
not
need to calculate the coeFcient.
(a). An exam has 20 questions worth 5 points each. Each problem recieves 0,1,2,3,4,5 points. How
many ways are there for a student to score
r
points? (Note that we do not care how many points
any specifc question recieves, just the number of questions that recieve 0 points, 1 point etc.)
Let
e
1
, ..., e
20
be the score on question 1-20, and we have
e
1
+
···
+
e
20
=
r
where 0
≤
e
i
≤
5. So
we want the coef of
x
r
in
g
(
x
) = (1 +
x
+
x
2
+
x
3
+
x
4
+
x
5
)
20
.
(b). How many ways are there to distribute
r
identical forks to 10 people so that each person
recieves either one or two forks?
Coef of
x
r
in
g
(
x
) = (
x
+
x
2
)
10
.
(c). The number of ways a team can score
r
points in a basketball game? (In basketball, any single
shot is worth either one, two, or three points. We are only interested in the number of each type
of shot made, not the order in which they were made.)
Let
e
1
, e
2
, e
3
be the number of shots that score 1,2,3 points. We have
e
1
+ 2
e
2
+ 3
e
3
=
r
where
0
≤
e
i
. So we want the coef of Coef of
x
r
in
g
(
x
) = (1 +
x
+
x
2
+
x
3
+
. . .
)(1 +
x
2
+
x
4
+
x
6
+
. . .
)(1 +
x
3
+
x
6
+
x
9
+
. . .
).
(d). In how many ways can we make change for