Probability and Permutations

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Key Questions

  • Permutations of items are arrangements of items.


    Examples

    All six permutations of #{a,b,c}# are:

    #{abc, acb, bac, bca, cab, cba}#

    Also, all 6 permutations of #{a,b,c}# chosen at 2 items at a time are

    #{ab, ba, ac, ca, bc, cb}#


    I hope that this was helpful.

  • Permutations are arrangements of items, so the number of permutations is the number of arrangements of items.

    Let #P(n,r)# denote the number of permutations of #n# items chosen #r# items at a time. #P(n,r)# can be found by

    #P(n,r)=n cdot (n-1) cdot (n-2)cdot cdots cdot(n-r+1)={n!}/{(n-r)!}#.


    Example

    How many 3-digit codes are possible if each digit is chosen from 0 through 9, and no digits are repeated.

    We can think of 3-digits codes as permutations of #10# digits chosen #3# digits at a time since no digits are repeated. So, we have

    #P(10,3)=10 cdot 9 cdot 8=720#.

    Hence, there are 720 possible 3-digit codes.


    I hope that this was helpful.

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  • It is important in permutations by the definition of the word "permutation". If we ignore order, then we are discussing / choosing "combinations".

    Given set #{a, b, c, d}#, if we count #ab# and #ba# as the same, then we are counting combinations.
    If they are counted as different, then we are counting permutation.

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