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Math Help - Find Number of Arrangements

  1. #1
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    Find Number of Arrangements

    Ten people line up for a photo, including you and your friend. How many line arrangements are there if the photographer insists on keeping 3 people in between you and your friend?
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    Quote Originally Posted by RTC1996 View Post
    Ten people line up for a photo, including you and your friend. How many line arrangements are there if the photographer insists on keeping 3 people in between you and your friend?
    Exactly three people, or at least three people?

    Also, what have you tried so far?
    Last edited by undefined; October 6th 2010 at 07:09 AM.
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    Hello, RTC1996!

    Ten people line up for a photo, including you and your friend.
    How many line arrangements are there if the photographer insists
    on keeping exactly 3 people in between you and your friend?

    There are 10 people: . A\text{ (you)},\;B\text{ (your friend}),\:\text{ and 8 Others.}


    You and your friend are spaced like this:

    . . A\:\_\:\_\:\_\:B\;\;\text{ or }\;\;B\:\_\:\_\:\_\:A . . 2 ways


    Select 3 of the Others and arrange them in the spaces: .
    . . There are: . _8P_3 \:=\:\dfrac{8!}{5!} \:=\:8\cdot7\cdot6 \:=\:336 ways.


    Now we have 6 "people" to arrange: . \boxed{AxyzB}\; \text{and 5 Others}
    . . There are: . 6! \,=\,720 ways.


    Therefore, there are: . 2\cdot 336\cdot720 \:=\:483,\!840 .line arrangements.
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    What do the variables xyz mean in the rectangle having AxyzB? There are 6 "people" to arrange and 5 others. Can you explain this a bit more to me? What kind of probability question are we playing with here? Is this conditional probability?
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    Quote Originally Posted by RTC1996 View Post
    What do the variables xyz mean in the rectangle having AxyzB? There are 6 "people" to arrange and 5 others. Can you explain this a bit more to me? What kind of probability question are we playing with here? Is this conditional probability?
    First, this is really nothing to do with probability.
    Oh yes, it could be a question from a probability course.
    But it is not probability: it is simply a counting question.

    Once we select the three people to stand between you and your friend, which forms a block of ONE. Now in affect we have six blocks
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    Quote Originally Posted by RTC1996 View Post
    What do the variables xyz mean in the rectangle having AxyzB? There are 6 "people" to arrange and 5 others. Can you explain this a bit more to me? What kind of probability question are we playing with here? Is this conditional probability?
    Your original question asked to count the number of possible lines for the given restriction.

    You could calculate the probability of 3 people being between you and your friend
    if the 10 people line up randomly.
    Nothing like that is being asked.

    You are asked to count arrangements.

    You and your friend can be in 6 different relative positions

    YxxxFxxxxx
    xYxxxFxxxx
    xxYxxxFxxx
    xxxYxxxFxx
    xxxxYxxxFx
    xxxxxYxxxF

    now you can swop places with your friend.
    All the other people can move about in the "x" positions.

    Soroban split it up to count them easily.
    Think of the "rectangle" as a "person" or single group moving in unison.
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