Root of the polynomial $x(x-1)(x-2)\cdots(x-K)=C$ Is there an analytic way to obtain the highest root of the polynomial 
$x(x-1)(x-2)\cdots(x-K)=C$ in terms of $K$ and $C$? The integer $K \ll x$ and the constant $C$ are known. 
The other way to ask the question is given $K$ and $C$ find $N$ such that $\binom{N}{K}=C$


Is there an analytic way to obtain the highest root of the polynomial $x(x-1)\cdots(x-K)\!=\!C$

Doubtful, since it is well-known that there is no general formula for the roots of a polynomial with degree greater than $4$. Also, the inverse of the factorial or falling factorial function is not known to possess a closed form.
