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Thread: How many ways could a class of 18 students divide into group

  1. #1
    Junior Member
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    Sep 2006
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    How many ways could a class of 18 students divide into group

    How many ways could a class of 18 students divide into groups of 3 students each?

    Have 18 students take 3 out, then do that 6 times?
    6(18P3)

    is that right?


    Thanks

  2. #2
    Elite Member
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    Jan 2005
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    Try [tex]\L\frac{{18!}}{{\left( {6!} \right)\left( {3!} \right)^6 }}[/tex]

    Can you explain why that works?

    EDIT:
    I first read it as three groups not three in a group.
    “A professor is someone who talks in someone else’s sleep”
    W.H. Auden

  3. #3
    Elite Member
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    Re: How many ways could a class of 18 students divide into g

    Hello, wind!

    How many ways could a class of 18 students divide into groups of 3 students each?

    I'm sure that pka meant: [tex]\L\:\frac{18!}{6!(3!)^6}[/tex]


    Let's baby-step through this problem . . .

    First, we'll number the groups, 1 to 6.

    For group #1, there are: [tex]\,{18 \choose 3}[/tex] ways.
    For group #2, there are: [tex]\,{15 \choose 3}[/tex] ways.
    For group #3, there are: [tex]\,{12 \choose 3}[/tex] ways.
    For group #4, there are: [tex]\,{9 \choose 3}[/tex] ways.
    For group #5, there are: [tex]\,{6 \choose 3}[/tex] ways.
    For group #6, there are: [tex]\,{3 \choose 3}[/tex] ways.

    Hence, there are: [tex]\L\:\frac{18!}{3!\sout{15!}}\cdot\frac{\sout{15!}} {3!\sout{12!}}\cdot\frac{\sout{12!}}{3!\sout{9!}}\ cdot\frac{\sout{9!}}{3!\sout{6!}}\cdot\frac{\sout{ 6!}}{3!\sout{3!}}\cdot\frac{\sout{3!}}{3!\sout{0!} } \:=\:\frac{18!}{(3!)^6}[/tex] ways.

    But the groups are not numbered; their order is not considered.

    Since our answer includes the [tex]6![/tex] different orders of the six groups,
    . . we must divide by [tex]6![/tex]

    Answer: [tex]\L\:\frac{18!}{6!(3!)^6}[/tex]

    I'm the other of the two guys who "do" homework.

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