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Math Help - Stupid formula, I get the question but not the formula

  1. #1
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    Stupid formula, I get the question but not the formula

    So here i am again, yes with another retarde post in most of your eyes but I come here because I need help

    Here is the question.

    Consider the name "john smith" with j letters for the first name and k letters for the second name and where the space between the name is not part of the name. Give a formula for the number of permutations if the first and second names are permutated independently

    How is the formula written to explain this question?
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  2. #2
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    Re: Stupid formula, I get the question but not the formula

    Do you understand what a "permutation" is? One permutation of "John" is "John", another is "oJhn", yet another is "nJoh". You can show that there are n! (n factorial) permutations of n distinct letters or other objects. Since "John" has 4 distinct letters, there are 4!= 24 permutations. Similarly, "Smith" has 4 distinct letters so there are also 4!= 24 permutations of that. Since any permutation of "John" can be used with any permutation of "Smith" there are (24)(24)= 576 different ways of writing those. If, in addition, you allow swapping the two names- that is "nJoh miSth" is one way, "miSth nJoh" is another- multiply that by 2: 1152.
    (If you did not require keeping the two names separate, there would be 8!= 40320 ways to permute the 8 letters.)

    More generally, if you have a "word" with k distinct symbols and another with j distinct symbols, there would be k! ways to permute the first, j! ways to permute the second, so (k!)(j!) ways to permute both. Again, if you allow swapping the two words, there would be 2(k!)(j!) ways.

    Note that this does NOT apply to a general "word with k letters" because those "letters" must all be distinct. If the word were, say, "hello", there are 5 letters but not 5!= 120 ways of permuting them because many of them will have only the "l"s swapped and would not be "different words". That is, if we write "heLlo", then Lhoel is one permutation, lhoeL is another. But with "hello", they are the same word. Because there are 2!= 2 ways to interchange the "l"s, there are 5!/2!= 120/2= 60 ways to permute the letters of "hello".
    Last edited by HallsofIvy; August 15th 2012 at 11:21 AM.
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  3. #3
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    Re: Stupid formula, I get the question but not the formula

    John has 4 letters but Smith had 5 letters I'm sure that was just a simple mistake on your part but they have to be permutated independently, I understan permutations like you explain but "Give a formula for the number of permutations if the first and second names are permutated independently" What is the formula? How is it written?
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  4. #4
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    Re: Stupid formula, I get the question but not the formula

    Yes, (4!)(5!). Sorry about that. You'd think I could count by now.

    If the first name has k distinct letters and the second name has j distinct letters then the two names can be permuted in (j!)(k!) ways as I said before.
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  5. #5
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    Re: Stupid formula, I get the question but not the formula

    Ok thank you, I thought it would be some complicated formular haha thanks anyway
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