Lesson 11

\[\begin{align*}\binom{-k}{n}&=\frac{(-k)(-k-1)\dotsb(-k-n+1)}{n!}=\\&=(-1)^{n}\frac{k\cdot (k+1)\dotsb(k+n-1)}{n!}=\\&=(-1)^{n}\binom{n+k-1}{n}\ .\end{align*}\] Therefore, \[\frac{1}{(1-x)^k}=\sum_{n=0}^{\infty}(-1)^{n}\binom{n+k-1}{n}(-x)^{n}= \sum_{n=0}^{\infty}\binom{n+k-1}{n}x^{n} \ .\]

The resulting formula agrees with what we already know. Indeed, \(\frac{1}{(1-x)^k}\) is the generating function for the number of ways to choose \(n\) objects when there are \(k\) different types of objects.