I am trying to prove the following identity, which I am sure is correct: $$ \sum_{m=0}^{K}(-1)^m{n \choose m}{(n-m)r-1 \choose n-1}=1, $$ where $K:=\left[n\frac{r-1}{r}\right]$ for some integer $r$, and $[x]$ denotes the largest integer contained in $x$.

In the special case $r=2$ this is identity (3.111) from H. Gould's book "Combinatorial identities" (1972). For $r=2$ it was first stated by B. C. Wong in Amer. Math. Monthly in May 1930 as an open question. I am wondering if someone has seen a proof of (or can prove) this identity for any $r$. The upper summation limit$\left[n\frac{r-1}{r}\right]$ also occurs in Gould's book in identity (3.113), so it is not entirely unheard of.