Discrete MathematicsLecture 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) = n2

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 = p1k1p2

k2…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 A1, A2, …, 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 x1 + x2 + x3 + x4 = 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(2n, n)

Binomial Formula

• (a + b)n = C(n, k)akbn-k

• Show that C(n, k) = 2n

• Show that (-1)kC(n, k) = 0

• Express kC(n, k)3k in the closed form