How to show that $2 ^\binom{N}{2} \sim \exp(\frac{N^2}{2}\ln(2))$ How can we show that:
$2 ^\binom{N}{2} \sim \exp(\frac{N^2}{2}\ln(2))$ 
for large N.

$\binom{N}{2}=N(N-1)/2\approx N^2/2$. So $2^{\binom{N}{2}}=\exp(\binom{N}{2}\ln(2))\approx \exp(\frac{N^2}{2}\ln(2))$. Note that $N^2$ is in the exponent, not outside as you have written.
