How can I prove that $\left|\sum_{i=0}^r (-1)^i \binom{a}{i} \binom{n-a}{r-i}\right| \leq \binom{n}{r}$?

Question

This is a conjecture:

How can I prove that

\begin{equation} \left|\sum_{i=0}^r (-1)^i \binom{a}{i} \binom{n-a}{r-i}\right| \leq \binom{n}{r} \end{equation}

for $0\leq a \leq n$, $0\leq r \leq n$ and $n,r,a \in \mathbb{N}$ ?

Answer 1

We have $$\left|\sum_{i=0}^{r}\left(-1\right)^{r}\dbinom{a}{i}\dbinom{n-a}{r-i}\right|\leq\sum_{i=0}^{r}\dbinom{a}{i}\dbinom{n-a}{r-i}=\dbinom{n}{r} $$ where the last identity follows from the Chu-Vandermonde identity.