Elements of Combinatorics

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Elements of Combinatorics
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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.

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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

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Permutations

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

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Permutations

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

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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.

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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.

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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"

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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!

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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

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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.

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1st Property of Combinations

• Theorem. If r and n are integers such that
  , then

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2nd Property of Combinations

• Theorem. If r and n are integers such that
  , then

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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)

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Pascals Triangle
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Pascals Triangle
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Pascals Triangle
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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?