201001 Math 222 Midterm 2 Practice Questions
1. How many Social Insurance Numbers (9 digit sequences) have each of the
digits 1, 2, 3 and 4 appearing at least once?
2. How many ways are there to arrange the
n
integers 1
,
2
,...,n
in a circle
so that
i
is never immediately followed by
i
+ 1 and
n
is never immedi
ately followed by 1? Arrangements that diﬀer by rotating the circle are
considered to be the same.
3. A group of 10 people agree to participate in a headshaving fundraiser. In
how many ways can they line up for a “before” shaving photo and (later)
an “after” shaving photo if no person occupies the same place in both
photos?
4. Find a recurrence relation and initial conditions for
a
n
, the number of se
quences of length
n
≥
0 with elements from
{
A,B,C,D,E
}
which contain
none of
AA,AB
and
BB
.
5. Find and solve a recurrence relation and initial conditions for
s
n
, the
number of sequences of length
n
≥
0 with elements from
{
a,b,c,
1
,
2
,
3
,
4
}
in which there are no consecutive numbers (identical or distinct).
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 Spring '11
 HuangJing
 Integers, Recurrence relation, initial conditions, Fibonacci number, Generating function

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