Lemma: For any increasing sequence of positive integers $a_{n}$, such that $lim_{n \to \infty} \dfrac{a_n}{n}=0$, the sequence $\dfrac{n}{a_n}$ contains all positive integers. This beautiful lemma can be found in PFTB. i will search it if anyone interests So since $S_n$ is a increasing sequence of positive integers and $lim_{n \to \infty} \dfrac{S_n}{n}=0$. According to the lemma above, $\dfrac{n}{S_n}$ contains all positive intergers Then exist m such that $m = S_m.2^a$ so $v_2(\binom{2^aS_m}{S_m})$ $ = s_2(S_m) + s_2((2^a-1)S_{m}) - s_2(2^aS_{m})$ $= s_2((2^a-1)S_{m}) >= 1$ so $\binom{2^aS_m}{S_m}$ is an even number which mean there exist infinitely many n.