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Math Help - permutation

  1. #1
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    permutation

    Find the number of words formed by permuting all the letters of the word EXERCISES .
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  2. #2
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    Quote Originally Posted by mathaddict View Post
    Find the number of words formed by permuting all the letters of the word EXERCISES .
     n ! = n \times (n-1) \times (n-2) \times (n-3) \times ... \times 1 where  n is the number of letters in the word.

    Assuming that by "words" you mean combinations of letters, rather than coherent words with literary meanings.
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  3. #3
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    Quote Originally Posted by mathaddict View Post
    Find the number of words formed by permuting all the letters of the word EXERCISES .
    Do you know that the number of arrangements of objects where a_1 are of one kind, a_2 are of another kind,... a_n are of the nth kind is \frac{(a_1 + a_2 + ... a_n)!}{a_1 ! a_2 ! ... a_n!}
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  4. #4
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    Hello, mathaddict!

    This is a permutation with some duplicated objects.
    There is a formula for this, but you can derive it yourself.


    Find the number of words formed by permuting all the letters of the word EXERCISES.

    If we had 9 different letters, there would be 9! possible words.

    But there are 3 identical E's.
    Switching them would not produce a new word.
    So our answer is too large by a factor of 3!

    . . (The number of ways 3 objects can be arranged.)

    And there are 2 identical S's.
    They can be switched in 2! ways without producing a new word.
    So our answer is also too large by a factor of 2!

    So the answer is: . \frac{9!}{3!\,2!} \:=\:30,\!240

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