Hola
$$ \sum_{0 \leq j <i \leq n} j{n \choose i}$$
This is:
[texx]\displaystyle\sum_{i=0}^n{}{n \choose i}\displaystyle\sum_{j=0}^{i-1}{}j=\displaystyle\sum_{i=0}^n{}{n \choose i}\dfrac{i(i-1)}{2}[/texx]
Now note that:
[texx]f(x)=(1+x)^n=\displaystyle\sum_{i=0}^n{}\displaystyle\binom{n}{i}x^i[/texx]
Compute [texx]f''(1)[/texx].
Best regards.