201001 Math 222 Sample Final Exam
1. Let
S
1
and
S
2
be sequences of 7 decimal digits. Suppose we say that
S
1
is
equivalent
to
S
2
if
S
1
is a rearrangement of
S
2
. For example 1231236 is equivalent to 2263311.
(a) How many sequences are equivalent to 9955500?
(b) What is the maximum possible size of a set of sequences, no two of which are
equivalent?
2. Give a combinatorial argument to show that
2
n
2
!
= 2
n
2
!
+
n
2
.
3. Prove that no matter how 51 different integers are selected from
{
1
,
2
, . . . ,
100
}
the
selections contains
a
and
b
such that
b
is a multiple of
a
.
4. How many ways are there to place 8 rings on the 4 fingers of your left hand if:
(a) The rings are identical.
(b) The rings are identical and at least one ring must be put on each finger.
(c) The rings are different and the order of rings on a finger is not important.
(d) The rings are different and the order of rings on a finger is important.
(e) The rings are different, the order of rings on a finger is important, and at least
one ring must be placed on each finger.
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 Spring '11
 HuangJing
 Math, Vertex, Recurrence relation, Fibonacci number, Rings, Generating function

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