Suppose det ⁡ ∑ k = 0 n k ∑ k = 0 n C k   n k 2 ∑ k = 0 n C k   n k &#8721…

As given $\left|\begin{array}{cc}\sum _{k=0}^{n}k& \sum _{k=0}^n{}_{ }{}^{n}C{}_k{k}^2\\ \sum _{k=0}^n{}_{ }{}^{n}C{}_{k}k& \sum _{k=0}^n{}_{ }{}^{n}C{}_k{3}^k\end{array}\right|=0$











$=4^n$
now $\left|\begin{array}{cc}\frac{n\left(n+1\right)}{2}& n\left(n+1\right)2^{n-2}\\ n{2}^{n-1}& 4^n\end{array}\right|=0$

$n\left(n+1\right)2^{2n-1}-n^2\left(n+1\right)2^{2n-3}=0$

$2^{2n-1}-n{2}^{2n-3}=0$
$n=4$
Now