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Combinatorics - Partitions

  1. Jun 15, 2008 #1

    LHC

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    How many ways can you place 10 identical balls into 3 identical boxes? Note: Up to two boxes may be empty.

    I approached this problem as:

    Let B represent ball
    Let 0 represent nothing (empty)

    |box wall| 0 0 B B B B B B B B B B |box wall|

    So, there must be two other box walls that must be inserted, and they can be inserted in these places:

    |box wall| 0 0 B B B B B B B B B B |box wall|
    ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^

    So, that would make [tex]_{11}C _{2}=55[/tex]. However, my teacher says it's supposed to be 66. Could someone please explain why? Thanks.
     
  2. jcsd
  3. Jun 15, 2008 #2

    It's a really interesting and hard question :smile:

    Here's how I approached (using yours):

    | _ 1 _ 2 _ 3 _ 4 _ 5 _ 6 _ 7 _ 8 _ 9 _ 10 _ |

    11C2 when you put two lines in those dashes but do not put them in same blank (so there will always be three or two boxes)
    + 11 when you put both of them together (only one box)
     
  4. Jun 15, 2008 #3

    LHC

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    Oh, I get it! Thanks for your help! =D
     
  5. Jun 15, 2008 #4
    oops.. I worded it wrong
    "(so there will always be three or two boxes)"**
    "(only one box or two boxes)"**

    oo well, you got it ;)
     
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