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**Plato** But of course they are different solutions. If those were three children and we are giving out 10 cookies then it would makes a real difference.

You are correct, there are $\displaystyle {12 \choose 2}$ ways to place ten indistinguishable objects into three different cells.

But in addition to the examples you gave that number also includes 0+0+10, 0+10+0, and 10+0+0.

Do you want to include the possibility of an empty cell?

It appears that the problem you really want to solve is to find the number of partitions if the integer 10 into three or less summands.