The Bose Einstein result is used to calculate the total number of combinations when sampling with replacement. Recall that in combinations order doesn't matter. Imagine that we are trying to count the number of ways to give 5 treats to our 3 cats. In this case, we are "sampling" the cats (not the treats) when we give a cat a treat. We're sampling with replacement because we can feed a cat a treat multiple times, and order in which we feed the cats doesn't really matter (although the cats might think otherwise). All that matters at the end is how many treats each cat received. Let's imagine we have three feeding stations, one for each cat. One way to feed the 5 treats to the 3 cats is to feed the first cat three treats and one each to the other two. This could be visualized like this: `| * * * | * | * |` Considering all of the ways to permute the diagram, we can move any of the dots (treats) and either of the bars (feeding stations) except for the outside two bars which must remain fixed. This gives us 7 spots for either a bar or dot, or $7!$ ways to permute them if they were all different. However, the treats are identical so we can divide out by the $5!$ ways to arrange the treats. Further, it's not legitimate to trade the feeding stations, so we divide out by the $2!$ ways to arrange the bars within themselves. This comes to $\frac{7!}{5!*2!}$ or $\binom{7}{5}$. Notice that $n + k - 1 = 7$ and $k = 5$. The $-1$ comes from the fact that we have one less movable bar than feeding stations (3 feeding stations result in 2 movable bars in the diagram). Thus this generalizes to $\binom{n + k - 1}{k}$ For example, the number of ways to create a four digit passcode in ascending order from the digits 0 through 9 can be calculated in this way. Here we are sampling with replacement (you can repeat the same number) but order doesn't matter. But you said they must be in ascending order so doesn't order matter!?! No, because what we're saying is actually that 1,2,3,4 is the same as 4,3,2,1 (because any set of numbers can be ordered and those with the same numbers will sort to the same values), so order doesn't matter. The answer would be $\binom{10 + 4 - 1}{4} = 715$.