Let $k\in\mathbb{N}$ and let $L_k = lcm(1,2,\cdots, k)$. Let $n\in\mathbb{N}$ with $n\ge k + L_k$. Prove that ${n\choose k}$ is divisible by $\prod_{i=0}^{k-1} \dfrac{n-i}{\gcd(n-i, L_k)}$ and conclude that ${n\choose k}$ has at least k distinct prime divisors.
I think Legendre's formula for $v_p(n!)$ for a prime p and positive integer n could be useful for finding stuff about ${n\choose k}$. The condition that $n\ge k + L_k$ ought to be useful in the problem statement, but what would happen if $n \ge k + L_k$? Seeing how things could go wrong might offer some insight as to how to tackle the problem. I think there are some useful properties about $L_k$ in regards to divisibility. If the terms in the product $\prod_{i=0}^{k-1} \dfrac{n-i}{\gcd(n-i, L_k)}$ were pairwise coprime, then it'd probably be easier to show that ${n\choose k}$ is a multiple of the product. But if we just take $k=3, L_k=6, n = 10,$ and the product is $10/2 \cdot 9/3 \cdot 8/2$, but $8/2=4$ and $10/2 = 5$ are not coprime.