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Prove or disprove that $\frac{(x+n)!}{x!}$ is not divisible by $n$ distinct primes where each prime is greater than $\frac{(x+n)e}{n}$
Is it true that for integers $n > e, x \ge n$, it is impossible for $\dfrac{(x+n)!}{x!}$ to be divisible by $n$ distinct primes where each prime is greater than $\dfrac{(x+n)e}{n}$?
Example:
$x=n=...
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