TITLE: What does $\binom{-n}{k}$ mean?
QUESTION [14 upvotes]: For positive integers $n$ and $k$, what is the meaning of $\binom{-n}{k}$?
Specifically, are there any combinatorial interpretations for it?

Edit: I just came across Daniel Loeb, Sets with a negative number of elements, Advances in Mathematics. 91 (1992), 64-74, which includes a combinatorial interpretation for $\binom{n}{k}$ for any $n,k \in \mathbb{Z}$ in theorem 5.2.

REPLY [3 votes]: For this answer it would probably be best if you're familiar with the Binomial distribution.
I like the interpretation through probability & statistics via the negative binomial distribution. Let's say we're watching a sequence of cointosses (let's call tossing a "heads" a success and "tails" a failure) where the probability of a success is independently $p$. Then the negative binomial distribution is a distribution over the waiting times till the $r$th success. 
In other words, what's the probability we'll have to wait $t$ tosses/trials to see $r$ successes, $p(t|r,p)$? 
Equivalently, by the $t-1$ trial we must have seen $r-1$ successes and $k=t-r+1$
failures. But then $t=k+r-1$, and so there are $\binom{k+r-1}{k}$ ways to observe $k$ failures and $r$ successes in $t$ trials. Since $r$ and
$k$ completely specify $t$, we can re-parameterize and write the
probability that the $r$th success occurs on the $(r+k)$th trial as
\begin{equation}
   p(k|r, p) = \binom{k+r-1}{k} p^r (1-p)^k
\end{equation}
Why the name negative binomial (and why is this an answer to your question)? Well, because 
\begin{equation}
  \binom{k+r-1}{k} = \frac{(k+r-1)_k}{k!} =
  \frac{(k+r-1)(k+r-2)\dots(r-1)}{k!} = (-1)^k \frac{(-r+1)\dots(-r
    -k+1)}{k} = (-1)^k \binom{-r}{k}
\end{equation}
Note that for integer $r$ the negative binomial distribution may be called the Pascal distribution.