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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) = 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 = p1k1p2k2…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 5person 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
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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
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(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