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Unformatted text preview: Sample Exam 3 Solution Ning SUN August 18, 2009 1. (20 pt) Derive a recurrence relation for a n , 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 a n and compute a 6 . There are two subcases based on the rst car: if a scion occupies the rst spot, then there are n- 1 spots left; if a hummer occupies the rst spot (it also occupies the second spot), then there are n- 2 spots left. So a n = a n- 1 + a n- 2 ; a = 1 , a 1 = 1 . This is a sequence of Fibonacci numbers. a 2 = 2 , a 3 = 3 , a 4 = 5 , a 5 = 8 , so a 6 = 13 . 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? De ne C: students who own cars, B: students who own bikes, M: students who own motorcycles. So N = 150 , N ( C ) = 83 , N ( B ) = 97 , N ( M ) = 28 . N ( C B ) = 127 , so N ( C B ) = N 4 + N 7 = N ( C ) + N ( B )- N ( C B ) = 53 . N ( C M ) = 97 , so N ( C M ) = N 6 + N 7 = N ( C )+ N ( M )- N ( C M ) = 14 . And N ( B M ) = N 5 + N 7 = 7 , N ( C M B ) = N 6 = 12 . So N 7 = 14- 12 = 2 , N 4 = ( N 4 + N 7 )- N 7 = 53- 2 = 51 , N 5 = ( N 5 + N 7 )- N 7 = 7- 2 = 5 . The requested is N ( B C M ) = N 2 = N ( B )- N 4- N 5- N 7 = 97- 51- 5- 2 = 39 . (b) How many students own a car and a bike but not a motorcycle? We want N 4 = 51 . 3. (20 pt) How many permutations of the 26 letters are there that contain NONE of the sequences INCH, LOST, or THIN? Constraint: (no INCH ) AND (no LOST ) AND (no THIN ). So we formulate it as an intersection of A 1 , A 2 and A 3 , where U (universe): all arrangements of the 26 letters; A 1 : all arrangements of the 26 letters containing INCH ; A 2 : all arrangements of the 26 letters containing LOST ; A 3 : all arrangements of the 26 letters containing THIN . The requested is: all arrangements of the 26 letters containing THIN ....
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- Spring '08