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Combinatorics - Partitions

  • Thread starter LHC
  • Start date
  • #1
LHC
24
0
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.
 

Answers and Replies

  • #2
378
2
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.

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)
 
  • #3
LHC
24
0
Oh, I get it! Thanks for your help! =D
 
  • #4
378
2
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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