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3. Counting

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Title: 3. Counting


1
3. Counting
  • Permutations
  • Combinations
  • Pigeonhole principle
  • Elements of Probability
  • Recurrence Relations

2
Permutations
  • Theorem 1. Suppose that two tasks T1 and T2 are
    to be performed in sequence. If T1 can be
    performed in n1 ways, and for each of these ways
    T2 can be performed in n2 ways, then the sequence
    T1T2 can be performed in n1n2 ways.
  • Theorem 1 is sometimes called the multiplication
    principle of counting.
  • Theorem 2. Suppose that tasks T1, T2,,Tk are to
    be performed in sequence. If T1 can be performed
    in n1 ways, and for each of these ways T2 can be
    performed in n2 ways, and for each of these n1n2
    ways of performing T1T2 in sequence, T3 can be
    performed in n3 ways, and so on, then the
    sequence T1T2 Tk can be performed in exactly
    n1n2nk ways.

3
Permutations
  • Problem 1. How many different sequences, each of
    length r, can be formed using elements from a set
    A if
  • elements in the sequence may be repeated?
  • all elements in the sequence must be distinct?
  • A sequence of r distinct elements of A is called
    a permutation of n objects taken r at a time,
    with An
  • When rn, the sequence is called a permutation of
    A, i.e. a distinct arrangement of the elements of
    a set A, with An, into sequence of length n

4
Permutations
  • Theorem 3. Let A be a set with n elements and
    1?r?n. Then the number of sequences of length r
    that can be formed from elements of A, allowing
    repetition, is nr.
  • Theorem 4. If 1?r?n, then nPr, the number of
    permutations of n objects taken r (distinct
    elements) at a time, is n(n-1)
    (n-r1)n!/(n-r)!
  • Theorem 5. The number of distinguishable
    permutations that can be formed from a collection
    of n objects where the first object appears k1
    times, the second objects k2 times, and so on, is
  • n!
  • k1!k2! kt!

5
Combinations
  • Problem 2. Let A be any set with n elements and
    1?r?n. How many different subsets of A are there,
    each with r elements.
  • the traditional name for an r-element subsets of
    an n-element set A is a combination of A, taken r
    at a time, where order does not matter
  • Theorem 1. Let A be a set with An, and let
    1?r?n. Then the number of combinations of the
    elements of A, taken r at a time, that is the
    number of r-element subsets of A is
  • n!
  • r!(n-r)!
  • nCr n!/r!(n-r)!, number of combinations of n
    objects taken r at a time.

6
Combinations
  • Theorem 2. Suppose k selections are to be made
    from n items without regards to order and that
    repeats are allowed, assuming at least k copies
    of each of the n items. The number of ways these
    selections can be made is (nk-1)Ck.
  • In general, when order matters, we count the
    number of sequences or permutations when order
    does not matter, we count the number of subsets
    or combinations

7
Pigeonhole Principle
  • The Pigeonhole Principle
  • If n pigeons are assigned to m pigeonholes, and
    mltn, then at least one pigeonhole contains two or
    more pigeons.
  • The principle provides an existence proof
  • The Extended pigeonhole Principle
  • If n pigeons are assigned to m pigeonholes, then
    one of the pigeonholes must contain at least
    ?(n-1)/m? 1 pigeons.

8
Elements of Probability
  • Probabilistic experiments do not yield exactly
    the same results when performed repeatedly
  • Deterministic experiments whose outcome is
    always the same
  • sample space A, a set consisting of all the
    outcomes of an experiment
  • event E a set of outcomes that satisfy some
    description
  • certain event A, impossible event ?
  • new events can be formed from events E and F
  • E?F occurs exactly when E or F occurs
  • E?F occurs if and only if both E and F occur
  • E occurs if and only if E does not occur
  • events E and F are mutually exclusive or disjoint
    if E?F, i.e. E and F cannot both occur at the
    same time
  • E1,E2,,Ek are mutually exclusive or disjoint if
    at most one of the events can occur on any given
    outcome of the experiment

9
Assigning Probabilities to Events
  • If event E has occurred nE times after n
    performances of the experiment, the fraction
  • fE nE/n is the frequency of occurrence of E in
    n trials
  • p(E), the probability of the event E, i.e. fE
    will tend to be p(E) when n becomes larger the
    frequencies of occurrence of event E
  • Rules for assigning probabilities
  • P1 0? p(E) ?1 for every event E in A
  • P2 p(A)1 and p(?)0
  • P3 p(E1?E2? ?Ek)p(E1)p(E2) p(Ek) whenever
    the events are mutually exclusive
  • If the probabilities are assigned to all events
    in such a way that P1, P2, P3 are always
    satisfied, then we have a probability space, and
    P1, P2, and P3 are called the axioms for a
    probability space.

10
Assigning Probabilities to Events
  • Let A be a probability space and is finite,
  • Ax1, x2, , xm, then each event xk,
    consisting of just one outcome, is called an
    elementary event
  • elementary probability corresponding to the
    outcome xk is pkp(xk).
  • elementary events are mutually exclusive
  • axioms of probability for elementary events
  • EP1 0? pk ?1 for all k
  • EP2 p1p2 pn1
  • If E is any event in A, and Exi1, xi2, ,
    xim, then Exi1? xi2? ? xim and
    p(E)pi1pi2 pim

11
Equally Likely Outcomes
  • Assume that all outcomes in a sample space A are
    equally likely to occur each has the same
    probability
  • random selection is to choose an object at random
    if all objects have an equal probability of being
    chosen
  • suppose An and these n outcomes are equally
    likely, then each elementary probability is 1/n,
    and for every event E, p(E)E/A
  • The expected value of an experiment is the sum of
    the value of each outcome times its probability
  • Roughly speaking, the expected value describes
    the average value for a large number of trials
  • Useful in analyzing the efficiency of algorithms,
    e.g. expected number of steps that the algorithm
    execute on an average run to give the output
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