Find the integer $k$ that satisfies the equality $$ \frac{1}{6 \choose k} - \frac{1}{7 \choose k} = \frac{16}{3 {8 \choose k}} $$
Attempt: My first thought was to directly plug in the definition for $n \choose k$ into the equation. But I ended up with a polynomial $2k^2 -33k + 112 = 0$, which does not give integer root. I've check with some friends, who got the same answer. Is there another way to solve this?