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Jonathan Thomas Guidry
CSE21
1.
(4 pts) The equation
x
1
+
x
2
+
x
3
+
x
4
=
20 has four distin
guishable variables.
(a) How many
nonnegative integer
solutions are there?
(b) Let the set of all the
nonnagative integer
solutions be
the sample space. What is the probability that all four variables
in a solution are positive?
(c) How many
integer
solutions are there in which no vari
able is less than

2?
(d) How many
integer
solutions are there in which no vari
able is greater than 8?
Hint:
These problems are variations of
Bars and Stars.
2.
(3 pts) Count each of the following
(a) the number of multisets of size 10 whose elements lie in
{
a
,
b
,
c
,
d
}
;
(b) the number of weakly increasing functions from
[
8
]
to
[
3
]
, i.e., the number of functions
f
:
[
8
]
→
[
3
]
such that
f
(
1
)
≤
f
(
2
)
≤ ··· ≤
f
(
8
)
.
(c) the number of strictly increasing functions from
[
4
]
to
[
7
]
, i.e., the number of functions
f
:
[
4
]
→
[
7
]
such that
f
(
1
)
<
f
(
2
)
<
···
<
f
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 Spring '07
 Graham

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