Discrete Mathematics Lecture 6

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Discrete MathematicsLecture 6
Alexander Bukharovich New York University
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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

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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?

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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

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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?

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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

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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 p1k1p2k2pmkm.
• 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?

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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?

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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?

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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?

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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?

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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?

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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?

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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

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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?

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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?

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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?

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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?

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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?

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Algebra of Combinations and Pascals Triangle

• The number of r-combinations from a set of n
  elements equals the number of (n
  r)-combinations from the same set.
• Pascals triangle C(n 1, r) C(n, r 1)
  C(n, r)

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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)

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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