Correlation between binomial coefficients

Question

I was wondering if there is any correlation between the binomial coefficients further than what wikipedia says. I found the following

\begin{equation} {13 \choose 6 }=2{13 \choose 4 } +{13 \choose 3 } \end{equation}

Is this random or is there a pattern?

Answer 1

Hint: $\dbinom{n}{k+1}=\dbinom{n}{k}\dfrac{n-k}{k+1}$

Hence,
$2\dbinom{13}{4}=2\dbinom{13}{3}\dfrac{10}{4}=5\dbinom{13}{3}$.

Finally,
$2\dbinom{13}{4}+\dbinom{13}{3}=6\dbinom{13}{3}=6\cdot\dfrac{13\cdot12\cdot11}{1\cdot2\cdot3}=\dfrac{10\cdot9\cdot8}{4\cdot5\cdot6}\cdot\dfrac{13\cdot12\cdot11}{1\cdot2\cdot3}=\boxed{\dbinom{13}{6}}$.