Coprimality with factorials

First. $n\mid n!$ since $n\le n$ and $n\ge1$, so $\gcd(n,n!)=n$.

Second. Any prime $p\mid n!$ satisfies $p\le n$, while any prime $p\mid n+1$ satisfies $p\le n+1$, but $p=n+1$ is possible only if $n+1$ is prime. If $p=n+1$ prime, then $p>n$, so $p\nmid n!$. If $p\le n$, then $p\mid n!$ but $p$ cannot divide $n+1$ unless $p=1$.

Conclusion. $\gcd(n,n!)=n$, $\gcd(n+1,n!)=1$.