This Is a Closed Book Exam; Only 1 Sheet (Both Sides) of Handwritten Notes Is Permitted

DiscreteDiscrete StructuresStructures forfor CSCS

Final Exam Date: Monday, May 7, 2007

Name: ______ID: ______

 This is a closed book exam; only 1 sheet (both sides) of handwritten notes is permitted.  You will have 135 minutes for this exam.  It will consist of 21 questions/problems worth 22 points each.

Instruction Work as many problems as possible. All problems have the same value. You may skip some problems, since you need to collect only 375 points out of 462. I suggest you try as many problems as possible, since for some of them you may get only partial credit. Please be concise and give well-written explanations/answers. Rambling or poorly written/organized explanations, which are difficult to follow, will receive less credit.

Problem 1 Problem 2 Problem 3 Problem 4 Problem 5 Problem 6 Problem 7

Problem 8 Problem 9 Problem 10 Problem 11 Problem 12 Problem 13 Problem 14

Problem 15 Problem 16 Problem 17 Problem 18 Problem 19 Problem 20 Problem 21 TOTAL

Problem#1: Prove that there are no solutions in integers x and y to the equation 2x2+5y2 =14.

Problem#2: Prove that there are infinitely many solutions in positive integers x, y and z to the equation x2+y2 =z2. n Problem#3: What is the term a7 of the sequence { an } if an equals to -(-3) ?

2 3 Problem#4: Compute the following double sum: Σ i=0,1,2 Σ j=0,1,2,3 i j . Problem#5: Show that 13+ 23+33+…+n3 = (n (n+1)/2)2 whenever n is a positive integer.

Problem#6: Find f(5) if f is defined recursively by f(0)=f(1)=1 and f(n+1)=f(n) / f(n-1) for n=2,3, …. Problem#7: Give a recursive definition of the set of positive integer powers of 2.

Problem#8: Verify that the program segment

If x< y then Min:=x Else Min:=y

٨min=x) v (x > y is correct with respect to the initial assertion T and the final assertion (x  y .(٨min=y Problem#9: A particular brand of shirt comes in 12 colors, has a male version and a female version, and comes in three sizes for each sex. How many different types of the shirt are made?

Problem#10: How many license plates can be made using either three letters followed by three digits or four letters followed by two digits? Problem#11: Suppose that there are nine students in a class at a small college. Show that the class must have at least three male students or at least seven female students.

Problem#12: A computer network consists of six computers. Each computer is directly connected to at least one of the other computers. Show that there are at lest two computers in the network that are directly connected to the same number of other computers. Problem#13: How many bit strings of length 12 contain at least three 1s or at least three 0s?

Problem#14: In how many different orders can five runners finish a race if no ties are allowed? Problem#15: What is the coefficient of x7 in (1+x)11?

Problem#16: Show that if n is a positive integer, then C (2n, 2) =2 C (n, 2) + n2 using a) a combinatorial argument, b) an algebraic manipulation. Problem#17: Find the next largest permutation in lexicographic order after the permutation 6 1 2 3 4 5.

Problem#18: What is the probability that a randomly selected day of the year 2007 is in May?

Problem#19: What is the probability that a five-card poker hand contains a flush, that is, five cards of the same suit? Problem#20: Find the probability of each outcome when a loaded die is rolled, if a 3 is twice as likely to appear as each of the other five numbers on the die.

Problem#21: What is the conditional probability that exactly four heads appear when a fair coin is flipped five times, given that the first flip came up tails?