If ${\left({}^{30}C_1\right)}^2+2{\left({}^{30}C_2\right)}^2+3{\left({}^{30}C_3\right)}^2..........30{\left({}^{30}C_{30}\right)}^2=\frac{\alpha 60!}{{\left(30!\right)}^2}$, then $\alpha$ is equal to 

Let,

$S=0{\left({}^{30}C_1\right)}^2+1{\left({}^{30}C_1\right)}^2+2{\left({}^{30}C_2\right)}^2+3{\left({}^{30}C_3\right)}^2..........30{\left({}^{30}C_{30}\right)}^2$

Now rewriting the above expression we get,

$S=30{\left({}^{30}C_{30}\right)}^2+29{\left({}^{30}C_{29}\right)}^2+28{\left({}^{30}C_{28}\right)}^2+27{\left({}^{30}C_{27}\right)}^2..........+0{\left({}^{30}C_0\right)}^2$

Now adding both the equation we get,

$2S=30\left[{\left({}^{30}C_1\right)}^2+{\left({}^{30}C_1\right)}^2+{\left({}^{30}C_2\right)}^2+{\left({}^{30}C_3\right)}^2..........{\left({}^{30}C_{30}\right)}^2\right]$

$\Rightarrow S=15\left[{}^{2\times 30}C_{30}\right]$

$\Rightarrow S=15\left[{}^{60}C_{30}\right]=15\times \frac{60!}{{\left(30!\right)}^2}$

Now comparing with $S=\frac{\alpha 60!}{{\left(30!\right)}^2}$,

We get $\alpha =15$