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Math Help - Permutation problems

  1. #1
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    Permutation problems

    ,

    I'm having a hard time solving these problems. How do you solve them?

    1) A family consisting of an old man, 6 adults and 4 children, is to be seated in a row for dinner. The children which to occupy two seats at each end and the old man refuses to have a child on either side of him. In how many ways can the seating arrangement be made for the dinner?

    2) Ten guests are to be seated in a row. Three of them are to be seated togather. Of the remaining seven, two do not wish to sit next to each other. Find the number of possible arrangements.

    3) The letters of the word ZENITH are written in all possible orders. If all the words are written in a dictionary, what is the rank of the word ZENITH?
    Last edited by saberteeth; October 26th 2009 at 02:23 AM.
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  2. #2
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    Quote Originally Posted by saberteeth View Post
    ,

    I'm having a hard time solving these problems. How do you solve them?

    1) A family consisting of an old man, 6 adults and 4 children, is to be seated in a row for dinner. The children which to occupy two seats at each end and the old man refuses to have a child on either side of him. In how many ways can the seating arrangement be made for the dinner?

    2) Ten guests are to be seated in a row. Three of them are to be seated togather. Of the remaining seven, two do not wish to sit next to each other. Find the number of possible arrangements.

    3) The letters of the word ZENITH are written in all possible orders. If all the words are written in a dictionary, what is the rank of the word ZENITH?
    You need to show that you have done some work on these.
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  3. #3
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    Quote Originally Posted by Plato View Post
    You need to show that you have done some work on these.
    Thanks for your reply Plato. I have spend hours trying to solve them but, i couldn't even figure out the starting point. This is why i am unable to show my workings
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  4. #4
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    Quote Originally Posted by saberteeth View Post
    I have spend hours trying to solve them but, i couldn't even figure out the starting point. This is why i am unable to show my workings.
    I find that incredible. How can anyone spend hours not knowing where to start?
    Maybe you need more help then we can give.
    Quote Originally Posted by saberteeth View Post
    3) The letters of the word ZENITH are written in all possible orders. If all the words are written in a dictionary, what is the rank of the word ZENITH?
    Can you at least tell us how many ways “ZENITH” can be rearranged?

    What is the last arrangement in a dictionary order?
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  5. #5
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    Quote Originally Posted by Plato View Post
    I find that incredible. How can anyone spend hours not knowing where to start?
    Maybe you need more help then we can give.

    Can you at least tell us how many ways “ZENITH” can be rearranged?

    What is the last arrangement in a dictionary order?
    That would be 6!
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  6. #6
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    Quote Originally Posted by saberteeth View Post
    That would be 6!
    Yes it is. Now answer these.
    What is the first arrangement in a dictionary order?
    What is the last arrangement in a dictionary order?
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  7. #7
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    Quote Originally Posted by Plato View Post
    Yes it is. Now answer these.
    What is the first arrangement in a dictionary order?
    What is the last arrangement in a dictionary order?
    The first arrangement is EHINTZ and the last one is ZTNIHE
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  8. #8
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    Quote Originally Posted by saberteeth View Post
    The first arrangement is EHINTZ and the last one is ZTNIHE
    Now there are 720 in the list.
    ZEHINT is #696.
    What number is ZENITH?
    You can list the other 23.
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  9. #9
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    Quote Originally Posted by Plato View Post
    Now there are 720 in the list.
    ZEHINT is #696.
    What number is ZENITH?
    You can list the other 23.
    This is the part that i am unable to understand. How did you know the order for ZEHINT is #696? Did you subtract 4! from all the possible arrangements? If yes, then what is the central idea behind finding the number of a particular arrangement?

    All this may sound stupid but, this is why i need your help as i am self-studying college math..
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  10. #10
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    Well I will try to help you, even though I think self-study of mathematics is not a good plan.
    There are 720 ways to arrange the letters “ZENITH”.
    There are 600 of those arrangements that do not begin with a Z.
    Therefore, in a dictionary ordering the arrangement “ZEHINT” is #601
    There are 24 arrangements that begin with a ZE.
    There are 12 arrangements that begin with a ZEH or ZEI.
    So the first that begins with a ZEN #613.
    There are 6 arrangements that begin with a ZEN.

    Can you finish?
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  11. #11
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    Quote Originally Posted by Plato View Post
    Well I will try to help you, even though I think self-study of mathematics is not a good plan.
    There are 720 ways to arrange the letters “ZENITH”.
    There are 600 of those arrangements that do not begin with a Z.
    Therefore, in a dictionary ordering the arrangement “ZEHINT” is #601
    There are 24 arrangements that begin with a ZE.
    There are 12 arrangements that begin with a ZEH or ZEI.
    So the first that begins with a ZEN #613.
    There are 6 arrangements that begin with a ZEN.

    Can you finish?
    Following your steps, i got my final answer, which is correct.

    ZENHIT # 613

    ZENHTI # 614

    ZENIHT # 615

    ZENITH # 616

    The only thing i did not understand is the statement "There are 12 arrangements that begin with a ZEH or ZEI" How did you get 12? Can you please elaborate?
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  12. #12
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    Ok, this took me few hours and i figured it out with the help of your steps. Thanks!
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