Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Midterm COMP 2804 March 2, 2016 1. What is the coefficient of x31 y 9 in the expansion of (4x − 6y)40 ? Answer: − 40 · 431 · 69 9 2. Let S = {1, 2, 3, 4, 5, 6, 7, 8}. You choose a uniformly random 6-element subset X of S. Thus, each 6-element subset of S has a probability of 1/ 86 of being X. Define the event A = “6 is an element of X” What is Pr(A)? Answer: 3/4 3. Let S be a collection of 6 positive and 8 negative numbers, where none of the numbers is 0. We draw 4 numbers from S, and multiply them (note that the numbers are drawn without replacement). What is the probability that the product of the four drawn numbers is positive. 8×(63)+6×(83) Answer: 1 − (144) 4. How many different strings of length 11 that can be formed by reordering the letters of the 11-letter word: M ISSISSAU GA 7 5 3 2 1 Answer: 11 4 2 2 1 1 1 5. How many solutions are there to the inequality x1 + x2 + x3 + x4 ≤ 20, where x1 ≥ 0, x2 ≥ 0, x3 ≥ 0, and x4 ≥ 0 are integers? Answer: 24 4 (See Theorem 3.9.2 in Michiel’s Notes) 6. The Board of Directors of ABC non-profit organization consists of a President, two VicePresidents, and a 7-person Advisory Board. The President cannot be Vice-President and cannot be on the Advisory Board. A Vice-President cannot be on the Advisory Board. Let n be the number of possible members to choose the Board of Directors for ABC, where n ≥ 10. In how many wayscan a Board of Directors be chosen? Answer: (n − 9) n2 n−2 7 1 7. Let S be a set of 20 elements and let x, y, and z be three distinct elements of S. What is the number of subsets of S that contain both x and y, but do not contain z? Answer: 217 8. For integers m ≥ 0, n ≥ 0, r ≥ 0, m ≥ r, and n ≥ r, the expression r X m n k r−k k=0 equals to Answer: m+n r 9. Let A be a set of 5 elements and let B be a set of 12 elements. How many one-to-one (i.e., injective) functions f : A → B are there? Answer: 8 · 9 · 10 · 11 · 12 10. For any integer n ≥ 2, let Sn be the number of bitstrings of length n in which the first two bits are the same. Which of the following is true? Answer: Sn = 2n−1 11. Consider strings of length 20 consisting of the characters a, b, and c. How many such strings are there that start with abc or end with bbb? Answer: 2 · 317 − 314 12. In any sequence of 50 distinct real numbers, there is always an increasing or a decreasing subsequence, whose length is at least: Answer: 8 13. An elementary school class contains 7 boys and 8 girls. If two students are selected at random to represent the class, what is the probability that they are of different sexes? Answer: 8/15 14. An elementary school class contains 7 boys and 8 girls. On two separate occasions a student is selected at random to represent the class. What is the probability that the two choices result in children of different sexes? Answer: 112/225 15. Consider the following recursive definition of a function f : N → N, where N is the set of natural numbers f (0) = 5 f (n) = 2f (n − 1) + 4, if n ≥ 1. What is f (n)? Answer: f (n) = 9 × 2n − 4 2 16. The function f : N → N is defined by f (0) = 5 f (n) = f (n − 1) + 2n − 3 for n ≥ 1 What is f (n)? Answer: f (n) = n2 − 2n + 5 17. Consider the following recursive algorithm Fib, which takes as input an integer n ≥ 0: Algorithm Fib(n): if n = 0 or n = 1 then f = n else f = Fib(n − 1) + Fib(n − 2) endif; return f When running Fib(10), how many calls are there to Fib(5)? Answer: 8 3 4 5