Studying thoroughly a physics matrix system I found (I'm pretty sure that I'm not committing errors) that for $P$ prime number and $n$ any integer $n<P$, we can decompose, $$\binom{P}{n}=P \ q$$ Where $q$ is an integer that depending on $P$ and $n$ will change values.
My intuition tell me that this is very related to Fermat's little theorem, but looking bibliography I can not find this concrete expression or something analogue anywhere. So I'm searching for someone that help me to find a place where this theorem or corollary is shown, or a way to prove it.
If $P$ is prime with the bound conditions, we have that ${P}\choose {n}$$=P \cdot \frac{(P-1)!}{n!(P-n)!}=Pq$. Obviously $P \mid Pq$, but only if $q \in \mathbb{Z}$.
As $p \in \mathbb{Z}$ , $Pq \in \mathbb{Z}$ , $P \cdot \frac{(P-1)!}{n!(P-n)!}=P \cdot \frac{(P-1)!\dots (P-n+1)}{k!}$, and $P$ coprime to all the factors of $k!$, $P$ has no contribution to make $Pq \in \mathbb{Z}$ which implies that $q$ must be an integer.