Consider the following identity $$\sum_{n=0}^{R-t}\binom{n+\ell}n\binom{R-\ell-n}{R-t-n}=\binom{R+1}{t+1}.\tag1$$ It is relatively easy to give an algebraic or mechanical proof of (1). But, I like to ask:
QUESTION. is there a combinatorial reason why the sum in (1) is independent of $\ell$?
(I am late to post a very similar answer to the already given one, yet I'd like to post it as well, since it differs in some detail)
Changing the summation index to $m=n+\ell$, the identity writes $$\sum_{m=\ell}^{R-t+\ell } {m\choose \ell}{R-m\choose t-\ell}={R+1\choose t+1}.$$
Given natural numbers $\ell\le t\le R$, and $m$, we may consider the class of those $(t+1)$-subsets $\{x_0<x_1<\dots<x_t\}$ of $\{0,1,\dots,R\}$ such that $x_\ell=m$: these are exactly ${m\choose \ell}{R-m\choose t-\ell}$ (indeed the $\ell$ elements $x_0,\dots, x_{\ell-1}$ can be chosen freely into $\{0,\dots, m-1\}$, and so can the $t-\ell$ elements $x_{\ell+1},\dots,x_t$ into $\{m+1,\dots,R\}$. These classes, for $ \ell\le m\le R-t+\ell $ form a partition of all $(t+1)$-subsets of $[R+1]$, whence the sum of their cardinality is independent of $\ell$ and the identity.
Well, here's a combinatorial proof of the identity anyway (but not a direct proof of independence of $\ell$).
Write $N = R - t$. Then the identity is $$\sum_{n=0}^N \binom{n + \ell}{n} \binom{R - n-\ell}{N - n} = \binom{R + 1}{N}.$$ The RHS is the number of $N$-sets $\{x_1 < \cdots < x_N\} \subset \{1, \dots, R+1\}$. Consider the last index $n$ such that $x_n \leq n + \ell$ (say $n = 0$ if there is no such index). Then $\{x_1 < \cdots < x_n\} \subset \{1, \dots, n+\ell\}$ and $\{x_{n+1} < \cdots < x_N\} \subset \{n+\ell+1, \dots, R+1\}$. The number of ways of making these choices is the LHS.
