Number of ways to interleave two ordered sequences. Suppose we have two finite, ordered sequences $x = (x_1,\dots,x_m)$ and $y = (y_1,\dots,y_n)$. How many ways can we create a new sequence of length $m+n$ from $x$ and $y$ so that the order of elements is preserved?
How many ways can we do this using $k$ ordered sequences instead of two?

This is the same as taking $m + n$ places, and deciding which $m$ of them get the $x_i$, i.e., $\binom{m + n}{m}$.
For $k$ ordered sequences, the basic idea is the same; you'll get an $k$-nomial coefficient.
