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Math Help - Counting problem

  1. #1
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    Counting problem

    A women wants to distribute 12 identical cookies to her 5 children of different ages.
    The youngest has to get at least 2 cookies while the rest at least 1.
    In how many way can she do it ?
    The answer is 210

    My unsuccessful attempt was dividing the problem
    first, we give 1 of the children 2 cookies and the rest 1 cookie.
    She can do that in 1 way.

    Then we are left with 12-6 = 6 cookies to distribute.
    We calculate "6 choose 5".

    Can some one help ?
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  2. #2
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    Quote Originally Posted by Hitman6267 View Post
    A women wants to distribute 12 identical cookies to her 5 children of different ages.
    The youngest has to get at least 2 cookies while the rest at least 1.
    In how many way can she do it ?
    The answer is 210
    There are \dbinom{N+K-1}{N} ways to distribute N identical objects to K different cells.
    Letís start by giving the youngest child two cookies.
    Then give each of the other four children one cookie each.
    That leaves six cookies to distribute any way the mother wants: \binom{6+5-1}{6}.
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  3. #3
    MHF Contributor Also sprach Zarathustra's Avatar
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    Solve the next linear equation:

    x_1 +x_2+x_3+x_4+x_5=12


    with following conditions:

    x_1>=2
    x_2,x_3,x_4,x_5>=1
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  4. #4
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    Hello, Hitman6267!

    A women wants to distribute 12 identical cookies to her 5 children of different ages.
    The youngest has to get at least 2 cookies while the rest get at least 1.
    In how many way can she do it?
    (The answer is 210)

    Place the 12 cookies in a row, leaving spaces between them.

    . . \circ\:\_\;\circ\;\_\;\circ\;\_\;\circ\;\_\;\circ\  ;\_\;\circ\;\_\;\circ \;\_\;\circ\;\_\;\circ\;\_\;\circ\;\_\;\circ\;\_\;  \;\circ


    Distribute 4 "dividers" among the 11 spaces.
    . . There are: . _{11}C_4 \:=\:{11\choose4} \:=\:330 ways.


    So that: . \circ\,\circ\,|\,\circ\,\circ\,\circ\,|\,\circ\, \circ\,|\,\circ\,|\,\circ\,\circ\,\circ\,\circ
    . . represents . \{2,3,2,1,4\} from the youngest to the oldest.

    And that: . \circ\,|\,\circ\,\circ\,|\,\circ\,\circ\, \circ\,|\,\circ\,\circ\,\circ\,\circ\,\circ\,|\,\c  irc
    . . represents \{1,2,3,5,1\}.



    But this includes distributions in which the youngest gets only one cookie.

    Very well, how many ways are there in which the youngest gets one cookie?

    We have: . \circ\,|\,\circ\,\_\,\circ\,\_\,\circ\,\_\, \circ\,\_\,\circ\,\_\,\circ\,\_\,\circ\,\_\,\circ\  ,\_\,\circ\,\_\,\circ\,\_\,\:\circ

    And we must distribute 3 dividers among the 10 spaces.
    . . There are: . _{10}C_3 \:=\:{10\choose3} \:=\:120 ways.


    Therefore, there are: . 330 - 120 \;=\;\boxed{210} ways in which the youngest
    . . gets at least 2 cookies and the others get at least one.

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