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.

Just use the definition!
$$\binom{N}{2} = \frac{N!}{2!(N-2)!} = \frac{N(N-1)}{2!} = \frac{N(N-1)}{2}$$
Now for $N\to\infty$ we can approximate $N(N-1) \approx N^2$ thence:
$$2^{\binom{N}{2}} \approx 2^{\frac{N^2}{2}}$$
Now we use the exponential representation, namely
$$a^b = e^{b\ln(a)}$$
And we get the desired result:
$$\large 2^{\frac{N^2}{2}} = e^{\frac{N^2}{2}\ln(2)}$$
