Explain why the count of positive / non-negative integer solutions to the equation $x_1 + x_2 + \cdots + x_k=n$ is equivalent to the case of putting $n$ indistinguishable balls into $k$ distinguishable boxes.
Let's take "putting $8$ instinguishable balls into $3$ distinct boxes" as an example to show that it is equivalent to $x_1+x_2+x_3=8$.
We arrange line $8$ indistinguishable balls up and divide them into three groups. The three groups are corresponding to three boxes which are labeled as $x_1$, $x_2$ and $x_3$, respectively.. Then, the numbers of balls in these three boxes are the value of $x_1$, $x_2$, and $x_3$, respectively. Swapping balls does not affect the count, hence does not change the values of $x_i$. However, swapping boxes will swapping the values of $x_i$ thus result in different solutions. Therefore, the boxes are distinguishable.
Hence, we conclude that these two models are equivalent.