How many ways to distribute 11 identical balls into 3 identical boxes with each box having 2 balls at least
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As each box must have at least two balls, let them first be filled with two balls each. Then, we have five balls left.
The ways in which five can be expresses as the sum of three non-negative integers (where order does not matter) are: $$5+0+0$$ $$4+1+0$$ $$3+2+0$$ $$3+1+1$$ $$2+2+1$$
Each represents a way in which the remaining balls can be put into the three boxes. Since the boxes are identical, each way will be counted once and the order doesn't matter.
So, number of ways$=5$.