If ∑ _ k = 1 ^ 31 (Ck 31)(Ck - 1 31) - ∑ _ k = 1 ^ 30 (Ck 30)(Ck -...

$\mathop{∑}_{k = 1}^{31} Ck 31 \cdot Ck - 1 31$

$= C1 31 \cdot C0 31 + C2 31 \cdot C1 31 + ⋯ + C31 31 \cdot C30 31$

$= C0 31 \cdot C30 31 + C1 31 \cdot C29 31 + ⋯ + C30 31 \cdot C0 31$ (using $Cr n = Cr - 1 n$)

$= C30 62$ (using $C0 n \cdot Cp m + C1 n \cdot Cp - 1 m + ⋯ + Cp n \cdot C0 m = Cp m + n$)

Similarly,

$\mathop{∑}_{k = 1}^{30}\left(Ck 30 \cdot Ck - 1 30\right) = C29 60$

$C30 62 - C29 60 = \dfrac{62 !}{30 ! 32 !} - \dfrac{60 !}{29 ! 31 !}$

$= \dfrac{60 !}{29 ! 31 !}\left\{\dfrac{62 \cdot 61}{30 \cdot 32} - 1\right\}$

$= \dfrac{60 !}{30 ! 31 !}\left(\dfrac{2822}{32}\right)$

So $\alpha = \dfrac{2822}{32}$

$∴ 16\alpha = 16 \times \dfrac{2822}{32} = 1411$