'm able to find an answer but I don't think it's the right one. I tested it for n=2 and it worked but not for n=3.Here's my answer:

$$C(n,2)+2×C(n,3)+3×C(n,4)+...+(n-1)×C(n,n)=[C(n,2)+C(n,3)+...+C(n,n)] + [0×C(n,2)+1×C(n,3)+2×C(n,4)+...+
(n-2)×C(n,n)] =
2^{(n-2)} + 2^{(n-2)}×(n-2)$$

I'm either forgetting something or it's just fully wrong. Sorry for the bad formatting and thank you in advance