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Permutations vs combinations

  1. Jan 6, 2009 #1
    What's the difference between these two:

    1) The number of permutations of n distinct objects taken r at a time is [tex]\frac{n!}{(n-r)!}[/tex]

    and

    2) The number of combinations of n distinct objects taken r at a time is [tex]\frac{n!}{r!(n-r)!}[/tex]

    ?
     
  2. jcsd
  3. Jan 6, 2009 #2

    statdad

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    Homework Helper

    Both ideas deal with counting the number of ways to make selections. For the formulas you have,

    Permutations
    * You have a collection of [tex] n [/tex] distinct items
    * You select [tex] r [/tex] of them without replacement
    * You are concerned with the order of selection

    Combinations
    * You have a collection of [tex] n [/tex] distinct items
    * You select [tex] r [/tex] of them without replacement
    * You are not concerned with the order of selection

    Suppose your set is [tex] \{a, b, c, d\}

    The number of permutations of 2 things taken from this group is [tex] 12 [/tex]. They are (order is first selected, second selected)
    a, b
    a, c
    a, d
    b, a
    b, c
    b, d
    c, a
    c, b
    c, d
    d, a
    d, b
    d, c

    The number of combinations of two things taken from this group is [tex] 6 [/tex]. They are
    a,b
    a,c
    a,d
    b,c
    b,d
    c,d

    Think this way: combinations count subsets - order is not important.
     
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