let $n$ an integer sufficiently large.

i'm looking for a "big" set $S$ of prime number $p$ satisfying $10.n \geq p > \sqrt{10n} $ and $p| C_{9.n}^{k-n} .C_{8n+k}^{8.n}$ for all integer $k$ satifying $ n \leq k \leq 10.n$ and $k \geq p $

that means I'm looking a havy condition $(C)$ over the $ p $ prime $10.n \geq p > \sqrt{10n} $ giving me for all $p$ prime satisfying $(C)$ we have

$\forall k$ integer such that $max(n,p) \leq k \leq 10.n$

$$ (I): v_p(C_{9.n}^{k-n} .C_{8n+k}^{8.n})=[\frac{8.n+k}{p}]-[\frac{k}{p}]-[\frac{8.n}{p}]+[\frac{9.n}{p}]-[\frac{k-n}{p}]-[\frac{10.n-k}{p}] \geq 1$$

where $[x]$ is the integer part of a real number $x$

( i think $(C) $ is of type $p \geq \sqrt{9.n}, \frac{a}{b}<\frac{n}{p}-[\frac{n}{p}]< \frac{c}{d} $for some integer $a,b,c,d $ that i coudn't find, wich is give us to the existence of an integer $h$ such that $p \in ]\displaystyle \frac{d.n}{c+d.h}, \displaystyle \frac{n.b}{a+r.h}[ $ )\

i prooved that $ \forall p \in]8.n,9.n[,$ i have $(I)$ there are many others prime number satisfying $(I)$ may be in $]\frac{8n}{2},\frac{9n}{2}[, ]\frac{8n}{3},\frac{9n}{3}[...$ i don't know how the condition... Any help please