In combinatorics, there is a formula "$n$ multichoose $k$", which is the way of making a multiset having $k$ elements choosing out of $n$ options. "$n$ multichoose $k$" is the same as "$(n+k-1)$ choose $k$". Why is that?
Let's say you have $N$ items (all alike for now) and $K-1$ vertical bars (all alike for now).
How many unique ways can you line up the $N$ items and the $K-1$ vertical bars?
Now pretend that everything to the left of the first bar is Type $1$, everything between the first and second bar is Type $2$ ... and everything to the right of the last $(K-1)$th bar is Type $N$.
Do you see the connection?