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sample exam3

# sample exam3 - AMS 301 Sample Exam 3 Summer 2009 Ning SUN...

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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 dierent 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? 1 ...
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