I've been reading 'Probability Theory : The Logic of Science' by E. T. Jaynes and am having trouble understanding equation 6.16 of the book.
Specifically, given the problem of drawing from an urn without replacement, the equation states that:
\begin{align} \sum\limits_{R=0}^N \binom{R}{r} \binom{N-R}{n-r} &= \binom{N+1}{n+1} \end{align}
Where given an urn of red and white balls, $N$ is the total number of balls, $R$ is the total number of red balls, $n$ is the number of balls we draw without replacement, and of those $r$ is the number of red balls drawn.
I can't seem to derive this equation myself. Any help would be greatly appreciated. :)
Both sides of the equation count the ways to pick $n+1$ elements from a set of $N+1$ elements (say, $\{0,\ldots,N\}$). This is clearly the case for the right-hand side. For the left-hand side, first pick $R$ as the $(r+1)$-th element, then pick the $r$ smaller elements from the $R$ elements less than $R$, and then pick the $n-r$ greater elements from the $N-R$ elements greater than $R$.