Explain why we cannot apply the cut-the-rope technique to count the non-negative integer solutions to the equation $$x_1 + x_2 + \cdots + x_k = n$$
For example, can we allow two cuts in the same interval thus to model one of the $x_i$ is zero?
One way to explain this is that we cannot translate this scenario (i.e. permits multiple cuts in the same interval) into a proper counting language.
For positive integer solutions, it is equivalent to "selecting $(k-1)$ out of $(n-1)$ available intervals to make one cut, order does not matter" which can be directly translate to the answer $C_{n-1}^{k-1}$. However, in the case of non-positive integer solutions, it is not possible to translate into the a counting language like this.