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17 Pages

### 52 Combinatorics

Course: CS 103, Fall 2009
School: Stanford
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Word Count: 5053

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#52 Handout November 14, 2008 CS103A Robert Plummer Combinatorics This handout presents in prose form many of the principles and examples discussed in class. Combinatorics is the study of counting, which is important in Computer Science in many ways: To understand the performance of algorithms, we need to count the steps they execute We also need to count the amount of memory used as algorithms execute...

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CS103A HO# 53 Slides-Combinatorics II11/14/08Sets What is |A B| ? A (A and B not necessarily disjoint) B AWhat is |A B C| ? B|A| = 8|B| = 6 C|A| + |B| = 14, which double counts the intersection |A B| = |A| + |B| - |A B| This is known
Stanford - CS - 103
CS103A HO #54 Slides-Combinatorics III, Functions11/17/08C(3,0) 1C(2,0)C(1,0) C(3,1) 3C(0,0) C(2,1)C(1,1) C(3,2) 3C(0,0) C(2,2) C(3,3) 1 C(3,0) C(4,0) 1 C(4,1) 4 C(1,0) C(2,0) C(3,1) C(2,1) C(3,2) C(4,2) 6 C(1,1) C(2,2) C(3,3) C(4,3) 4
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Handout #55 November 17, 2008CS103A Robert PlummerFunctionsAs used in ordinary language, the word function indicates dependence of one quantity on another. More specifically, suppose two sets of objects are given: set A and set B; and suppose th
Stanford - CS - 103
Handout #56 November 17, 2008CS103A Robert PlummerProblem Set #8 Solutions1) The formula is n(n+1). Proof: We will show that property P(n) holds for all integers n &gt; 0 using the weak mathematical induction. Here, property P(n) means that 2 + 4 +
Stanford - CS - 103
CS103AHO #57Functions II11/19/08InverseA function f: X Y is called invertible if and only if there exists a function g : Y X such that y = f(x) x = g(y) for all x X and for all y Y. We call g the inverse of f and write g = f-1. f y0 f y
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Handout #58 November 19, 2008CS103A Robert PlummerProblem Set #10Due: Wed., December 3 (Turn in by noon, Dec. 5 at Bob's office for one late day) Combinatoric problems can be very misleading. They seem very simple, but may turn out to be surpris
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CS 103A HO #60 CombinatoricsReview Session and Extra Problems11-21-20081 (Online). Five couples go to the movies together and sit in a row of ten seats. In how many ways can the 10 people be arranged if: (a) They may sit in any order. Solution Th
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Handout #63 December 3, 2008CS103A Robert PlummerRevised Combinatorics ChartHere is a revised version of the chart for combinatorics formulas. It correctly shows that when we are selecting r objects from a set of n objects with repetition allowe
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Handout #64 December 3, 2008CS103A Robert PlummerPractice FinalExam Rules: 1) You have 3 hours to complete this exam. 2) This is an open-note exam no textbooks allowed. Handouts, problem sets and solutions are allowed. You may also bring a one-
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Handout #65 December 3, 2008CS103A Robert PlummerProblem Set #9 Solutions1) a1 = 6, an+1 = an + 6 for n 1 2) P(n): Base Case: P(0) asserts that x =0i =0nx =i1 xn+11 x1 1xSince both sides are 1, this is true. 1 xk+1 1-xIndu
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Handout 66 December 7, 2008CS103A Robert PlummerProblem Set #10 Solutions1) Product rule: 30 * 12 * 4 *6 2) a) The two kinds of numbers don't overlap (e.g., if there are exactly four 0's then there are four 1's, not three), so it's the number of
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Handout #67 December 7, 2008CS103A Robert PlummerPractice Final Solutions1. Number Theory (20 points) (a) (5 points) Prove or Disprove the following: The total number of distinct positive divisors of any positive integer is even. Counterexample:
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Handout #68 December 7, 2008CS103A Robert PlummerFinal Exam Review SessionNumber Theory 1. (20 points) Prove that for all positive integers n: 6 | (n (n + 1) (2n +1). Induction is not required for this proof (but is allowed).Induction 2. (20 p
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50.7000 44.0920 54.0040 57.7296 49.5032 60.1744 49.7360 45.6200 71.7936 54.0888 48.1592 62.2320 57.5096 70.3368 65.3608 46.7840 57.8912 61.2024 64.5216 69.0184 59.7592 63.4120 68.8832 63.8432 66.3
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2.1102 -0.5571 0.6350 -0.7500 -2.4973 5.4865 -7.0362 4.9175 -3.5334 4.2431 -2.7678 1.0740 -0.9709 1.5055 -2.8513 2.2936 -0.3202 -1.0711 0.5841 2.1127 -3.7291 1.9311 -0.1280 0.5305 -1.5
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3.3937 1.6444 2.7582 1.3890 2.5195 2.1712 2.1593 2.9381 1.6889 3.4285 2.1186 3.9256 3.3147 1.0789 2.9780 1.5067 2.6848 2.0754 2.5234 4.0964 2.4909 2.5986 2.8088 1.0751 2.1
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22.6710 20.7000 23.3200 21.8260 22.2990 22.3850 21.5970 23.1140 22.7030 24.3620 25.5700 27.4760 26.9680 24.1590 28.5170 27.1830 28.3320 27.5760 29.1890 31.3820 30.6540 33.7650 34.2700 36.6700 37.3
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60.0000 52.6920 54.5960 59.5256 55.5288 61.5032 56.3392 46.3592 61.8528 58.9416 49.4008 53.2328 51.6424 62.1792 67.0400 52.6240 48.6152 47.2176 48.4392 53.4576 48.5168 46.5288 49.3120 46.3552 45.
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50.7000 76.0432 71.9192 63.2144 52.0880 59.3992 59.3992 51.5712 56.7528 67.3248 75.7672 75.6616 78.5208 67.6472 78.1784 79.0016 68.4512 71.0920 74.5408 63.3224 74.9080 78.8224 81.6776 88.8248 91.8