How many ways can you partition 10 identical balls into 3 identical boxes?

[LHC]

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 $$_{11}C _{2}=55$$. However, my teacher says it's supposed to be 66. Could someone please explain why? Thanks.

 

[rootX]

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 $$_{11}C _{2}=55$$. However, my teacher says it's supposed to be 66. Could someone please explain why? Thanks.

It's a really interesting and hard question

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)

 

[LHC]

Oh, I get it! Thanks for your help! =D

 

[rootX]

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 ;)