If (C1 30) ^ 2 + 2 (C2 30) ^ 2 + 3 (C3 30) ^ 2 . . . . . . . . . ....

Let,

$S = 0{\left(C1 30\right)}^{2} + 1{\left(C1 30\right)}^{2} + 2{\left(C2 30\right)}^{2} + 3{\left(C3 30\right)}^{2} . . . . . . . . . . 30{\left(C30 30\right)}^{2}$

Now rewriting the above expression we get,

$S = 30{\left(C30 30\right)}^{2} + 29{\left(C29 30\right)}^{2} + 28{\left(C28 30\right)}^{2} + 27{\left(C27 30\right)}^{2} . . . . . . . . . . + 0{\left(C0 30\right)}^{2}$

Now adding both the equation we get,

$2S = 30\left[{\left(C1 30\right)}^{2} + {\left(C1 30\right)}^{2} + {\left(C2 30\right)}^{2} + {\left(C3 30\right)}^{2} . . . . . . . . . . {\left(C30 30\right)}^{2}\right]$

$⇒ S = 15\left[C30 2 \times 30\right]$

$⇒ S = 15\left[C30 60\right] = 15 \times \dfrac{60 !}{{\left(30 !\right)}^{2}}$

Now comparing with $S = \dfrac{\alpha60 !}{{\left(30 !\right)}^{2}}$,

We get $\alpha = 15$