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Elements of Combinatorics

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Let students take during this semester Calculus (C), Physics (P), and Discrete Mathematics(D) classes, ... Basic Concepts of Discrete Probability Author: Igor ... – PowerPoint PPT presentation

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Title: Elements of Combinatorics


1
Elements of Combinatorics
2
Permutations
  • (Weak Definition) A permutation is usually
    understood to be a sequence containing each
    element from a finite set once, and only once.
  • The concept of sequence is distinct from that of
    a set, in that the elements of a sequence appear
    in some order the sequence has a first element
    (unless it is empty), a second element (unless
    its length is less than 2), and so on. In
    contrast, the elements in a set have no order
    1, 2, 3 and 3, 2, 1 are different ways to
    denote the same set.

3
Permutations
  • (General Definition) A permutation is an ordered
    sequence of elements selected from a given finite
    set, without repetitions, and not necessarily
    using all elements of the given set.
  • For example, given the set of letters
    C, E, G, I, N, R, some permutations are
    ICE, RING, RICE, NICER, REIGN and CRINGE

4
Permutations
  • The total number of different permutations of n
    elements of a set with the cardinality n is

5
Permutations
  • The number of different (ordered) permutations
    (arrangements) of r elements selected from n is

6
Combinatorics where we need it?
  • For example, if students have today Calculus (C),
    Physics (P), and Discrete Mathematics (D)
    classes. How we can calculate the probability
    that D is the first class?
  • The following 6 arrangements are possible CPD,
    CDP, PCD, PDC, DCP, DPC. Two of them are
    desirable DCP and DPC. Thus, if all events are
    equiprobable, then the probability is 2/61/3.

7
The number of subsets of a set
  • Theorem. If n is any nonnegative integer, then a
    set of the cardinality n (a set with n elements)
    has exactly 2n subsets.

8
Combinations
  • A combination is an un-ordered collection of
    distinct elements, usually of a prescribed size
    and taken from a given set.
  • Given a set S, a combination of elements of S is
    just a subset of S, where, as always for
    (sub)sets the order of the elements is not taken
    into account (two lists with the same elements in
    different orders are considered to be the same
    combination). Also, as always for (sub)sets, no
    elements can be repeated more than once in a
    combination this is often referred to as a
    "collection without repetition"

9
Combinations
  • The number of different (not ordered)
    combinations of r elements selected from n is the
    number of all possible permutations
    (arrangements) of r objects
    selected from n
    divided by the number of all possible
    permutations of r objects r!

10
The number of subsets of a set
  • Theorem. Let S be a set containing n elements,
    where n is a nonnegative integer. If r is an
    integer such that , then the
    number of subsets of S containing exactly r
    elements is

11
Combinatorics where we need it?
  • Let students take during this semester Calculus
    (C), Physics (P), and Discrete Mathematics(D)
    classes, two classes/day. How we can calculate
    the probability that D and P are taken at the
    same day?
  • There are 3 different combinations of 2 objects
    selected from 3 (CPPC), (CDDC), (DPPD). One
    of them is desirable (DPPD). Thus, the
    probability is 1/3.

12
1st Property of Combinations
  • Theorem. If r and n are integers such that
    , then

13
2nd Property of Combinations
  • Theorem. If r and n are integers such that
    , then

14
Binomial Theorem
  • Binomial Theorem (I. Newton). Let x and y be the
    variables, and n is a nonnegative integer. Then
    are the
    coefficients of the binomial decomposition
    (binomial coefficients)

15
Pascals Triangle
16
Pascals Triangle
17
Pascals Triangle
18
Combinatorics where we need it?
  • Tournament problem. Suppose there are n chess
    players and they participate in the tournament,
    where everybody has to play with all other
    participants exactly 1 time. How many parties
    will be played in this tournament?
  • How many parties will be played if each
    participant has to play with others twice?
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