Discrete Mathematics Lecture 6 Alexander Bukharovich New York

Discrete Mathematics Lecture 6 Alexander Bukharovich New York University

Counting and Probability • Coin tossing • Random process • Sample space is the set of all possible outcomes of a random process. An event is a subset of a sample space • Probability of an event is the ratio between the number of outcomes that satisfy the event to the total number of possible outcomes • Rolling a pair of dice and card deck as sample random processes

Possibility Trees • Teams A and B are to play each other repeatedly until one wins two games in a row or a total three games. – What is the probability that five games will be needed to determine the winner? • Suppose there are 4 I/O units and 3 CPUs. In how many ways can I/Os and CPUs be attached to each other when there are no restrictions?

Multiplication Rule • Multiplication rule: if an operation consists of k steps each of which can be performed in ni ways (i = 1, 2, …, k), then the entire operation can be performed in ni ways. • Number of PINs • Number of elements in a Cartesian product • Number of PINs without repetition • Number of Input/Output tables for a circuit with n input signals • Number of iterations in nested loops

Multiplication Rule • Three officers – a president, a treasurer and a secretary are to be chosen from four people: Alice, Bob, Cindy and Dan. Alice cannot be a president, Either Cindy or Dan must be a secretary. How many ways can the officers be chosen?

Permutations • A permutation of a set of objects is an ordering of these objects • The number of permutations of a set of n objects is n! • An r-permutation of a set of n elements is an ordered selection of r elements taken from a set of n elements: P(n, r) • P(n, r) = n! / (n – r)! • Show that P(n, 2) + P(n, 1) = n 2

Exercises • How many odd integers are there from 10 through 99 have distinct digits? • How many numbers from 1 through 99999 contain exactly one each of the digits 2, 3, 4, and 5? • Let n = p 1 k 1 p 2 k 2…pmkm. – In how many ways can n be written as a product of two mutually prime factors? – How many divisors does n have? – What is the smallest integer with exactly 12 divisors?

Addition Rule • If a finite set A is a union of k mutually disjoint sets A 1, A 2, …, Ak, then n(A) = n(Ai) • Number of words of length no more than 3 • Number of integers divisible by 5 • If A is a finite set and B is its subset, then n(A – B) = n(A) – n(B) • How many students are needed so that the probability of two of them having the same birthday equals 0. 5?

Inclusion/Exclusion Rule • n(A B) = n(A) + n(B) – n(A B) • Derive the above rule for 3 sets • How many integers from 1 through 1000 are multiples of 3 or multiples of 5? • How many integers from 1 through 1000 are neither multiples of 3 nor multiples of 5?

Exercises • Suppose that out of 50 students, 30 know Pascal, 18 know Fortran, 26 know Cobol, 9 know both Pascal and Fortran, 16 know both Pascal and Cobol, 8 know Fortran and Cobol and 47 know at least one programming language. – How many students know none of the three languages? – How many students know all three languages – How many students know exactly 2 languages?

Exercises • Calculator has an eight-digit display and a decimal point which can be before, after or in between digits. The calculator can also display a minus sign for negative numbers. How many different numbers can the calculator display? • A combination lock requires three selections of numbers from 1 to 39. How many combinations are possible if the same number cannot be used for adjacent selections?

Exercises • How many integers from 1 to 100000 contain the digit 6 exactly once / at least once? • What is a probability that a random number from 1 to 100000 will contain two or more occurrences of digit 6? • 6 new employees, 2 of whom are married are assigned 6 desks, which are lined up in a row. What is the probability that the married couple will have non-adjacent desks?

Exercises • Consider strings of length n over the set {a, b, c, d}: – How many such strings contain at least one pair of consecutive characters that are the same? – If a string of length 10 is chosen at random, what is the probability that it contains at least on pair of consecutive characters that are the same? • How many permutations of abcde are there in which the first character is a, b, or c and the last character is c, d, or e? • How many integers from 1 through 999999 contain each of the digits 1, 2, and 3 at least once?

Combinations • An r-combination of a set of n elements is a subset of r elements: C(n, r) • Permutation is an ordered selection, combination is an unordered selection • Quantitative relationship between permutations and combinations: P(n, r) = C(n, r) * r! • Permutations of a set with repeated elements • Double counting

Team Selection Problems • There are 12 people, 5 men and 7 women, to work on a project: – How many 5 -person teams can be chosen? – If two people insist on working together (or not working at all), how many 5 -person teams can be chosen? – If two people insist on not working together, how many 5 person teams can be chosen? – How many 5 -person teams consist of 3 men and 2 women? – How many 5 -person teams contain at least 1 man? – How many 5 -person teams contain at most 1 man?

Poker Problems • • • What is a probability to contain one pair? What is a probability to contain two pairs? What is a probability to contain a triple? What is a probability to contain straight? What is a probability to contain flush? What is a probability to contain full house? What is a probability to contain caret? What is a probability to contain straight flush? What is a probability to contain royal flush?

Exercises • An instructor gives an exam with 14 questions. Students are allowed to choose any 10 of them to answer: – Suppose 6 questions require proof and 8 do not: • How many groups of 10 questions contain 4 that require a proof and 6 that do not? • How many groups of 10 questions contain at least one that require a proof? • How many groups of 10 questions contain at most 3 that require a proof? • A student council consists of 3 freshmen, 4 sophomores, 3 juniors and 5 seniors. How many committees of eight members contain at least one member from each class?

Combinations with Repetition • An r-combination with repetition allowed is an unordered selection of elements where some elements can be repeated • The number of r-combinations with repetition allowed from a set of n elements is C(r + n – 1, r) • How many monotone triples exist in a set of n elements?

Integral Equations • How many non-negative integral solutions are there to the equation x 1 + x 2 + x 3 + x 4 = 10? • How many positive integral solutions are there for the above equation?

Algebra of Combinations and Pascal’s Triangle • The number of r-combinations from a set of n elements equals the number of (n – r)combinations from the same set. • Pascal’s triangle: C(n + 1, r) = C(n, r – 1) + C(n, r)

Exercises • Show that: 1 * 2 + 2 * 3 + n * (n + 1) = 2 * C(n + 2, 3) • Prove that C(n, 0)2 + C(n, 1)2 + … + C(n, n)2 = C(2 n, n)

Binomial Formula • • (a + b)n = C(n, k)akbn-k Show that C(n, k) = 2 n Show that (-1)k. C(n, k) = 0 Express k. C(n, k)3 k in the closed form