Unformatted text preview: AMS 301 Sample Exam 3
Summer 2009, Ning SUN
August 16, 2009 1. (20 pt) Derive a recurrence relation for an , the number of sequences of cars that can be parked
in a line of n spots if the only possible cars are scions and hummers, each scion requires just
1 spot and each hummer requires 2 spots (empty spots are not allowed). Write down an and
compute a6 .
2. (20 pt) A survey of 150 college students reveals that 83 own cars, 97 own bikes, 28 own
motorcycles, 127 own a car or a bike, 97 own a car or a motorcycle, 7 own a bike and a
motorcycle, and 12 own a car and a motorcycle but not a bike.
(a) How many students own just a bike?
(b) How many students own a car and a bike but not a motorcycle?
3. (20 pt) How many permutations of the 26 letters are there that contain NONE of the sequences
INCH, LOST, or THIN?
4. (20 pt) How many ways can a child take 12 pieces of candy, out of 4 types of candy, so that
the child does not take exactly two pieces of any type of candy?
5. (20 pt) Given the constraints on the matchings of 5 men (Rows) and 5 women (columns) in
the following gure:
(a) Find the rook polynomial.
(b) Give an expression for the number of matchings. 6. (30 pt) Derive a recurrence relation for an , the number of ways to give away n dollars:
(a) if each successive day you give away $1 or $2 or $3.
(b) if you cannot give away $1 one day, then $1 the next day.
(c) if you cannot give away $1 one day, then $1 the next day followed by $2 the third day.
7. (30 pt) The Bernstains, Hendersons, and Smiths each have 5 children. If the 15 children of
these three families camp out in ve dierent tents, where each tent holds 3 children, and the
15 children are randomly assigned to the ve tents, How many ways are there to do this so
that every family has at least two of its children in the same tent?
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- Spring '08
- Automobile, Recurrence relation, Motorcycle, Bicycle, Hendersons