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15b_Loehr - MATH 15B REVIEW PROBLEMS December 2000 1 MATH...

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MATH 15B REVIEW PROBLEMS December 2000 1 MATH 15B REVIEW PROBLEMS Disclaimer: These problems were written by me [Nick Loehr] and represent types of problems that can be solved based on the material Dr. Bender covered in 15B. The subject matter and difficulty level of these questions do not necessarily correspond to what you will see on the final exam. However, I’ve tried to design problems that cover lots of the topics mentioned in the course. The problems are divided into two groups. The reasonable problems are intended to be easy or medium-level in difficulty. Some of the interesting problems may be harder or trickier, but should be solvable based on what’s been talked about in the course. If you want even more difficult problems, just consult the course reader. Look on the course web page for a list of topics that Dr. Bender decided to exclude from the final exam after I wrote most of these review problems. I. Reasonable Problems. 1. How many four-digit numbers (with no leading zeroes) are multiples of 5 AND have all digits distinct? 2. Define a “valid word” to be any string of letters (from the 26 letters A through Z, repetitions allowed) with length less than five and containing at least one vowel (A, E, I, O, U). How many valid words are there? 3. How many functions f : { 1 , 2 , ..., 100 } → { 1 , 2 , ..., 100 } are there with the property that f (1) = 3 and f (47) = 1 and f (19) = 100 and f (2) = 2 and f (50) < 4? What’s the answer if f is also required to be a permutation? 4. An urn contains three balls labelled A, B, and C. Ball A has radius 1, ball B has radius 2, and ball C has radius 3. One ball is drawn from the urn. Assume that the probability of drawing a ball is proportional to its volume . (a) Find P(ball A is drawn), P(ball B is drawn), and P(ball C is drawn). (b) Let X be the radius of the drawn ball. Compute E [ X ] and Var( X ). 5. Solve these recurrences. Verify that your solutions are correct by induction. (a) f n +2 = - 6 f n +1 - 9 f n for n 0; f 0 = 2 and f 1 = 0. (b) f n +2 + f n = 0 for n 1; f 1 = f 2 = 1. 6. Draw a decision tree to make an alphabetical list of all four-letter words containing only the letters A, T, and H, and such that: (i) Two A’s are never adjacent. (ii) An H not at the end of the word is always immediately followed by an A. (iii) There are never three consecutive T’s. Based on this tree, what is the rank of the word HATH? 7. Let H be the (simple) graph with vertex set { 1 , 2 , 3 , 4 , 5 } and no edges. How many different subgraphs does H have? How many connected components does H have?
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