Product of divisors

$240=3\times5\times2^4$

By Tao $240$ has $2\times2\times5=20$ divisors

Now each $d_i$ has a $d_k$ such that $d_i\times d_k=240$ thus product of all divisors of $240$ is $\sqrt{240^{20}}={240}^{10}=2^{40}\times5^{10}\times3^{10}$

$\alpha=40$

$\beta=10$

$\gamma=10$ .

Finally

$\frac { \alpha +\beta +\gamma }{ 6}=\frac{40+10+10}{6}=\boxed{10}$