Solving equation involving binomial function Solve for $x$ in terms of $i$ and $j$:
$$
\binom{x}{i} = j
$$
where $x$ is Real; $i$ and $j$ are Integers: $x \geqslant i$, $i \geqslant1$, $j \geqslant 0$.
I came across this problem while trying to unrank combinations $\binom{n}{k}$, pp. 51, Algorithm 2.12. I tried the following:
$$
Minimize \left | \binom{x}{i}-j \right | 
$$
$$
s.t. x\geqslant i
$$
A plot of $|\binom{x}{i}-j|$ for $i=20$ and $j=10$ is the following:

I ignore if there is a closed-form for the real solutions of $x$ in terms of $i$ and $j$.

$\displaystyle{x\choose i}$ is a polynomial of degree i in x. For $i<5$, we have the linear, quadratic, cubic, and quartic formulas for roots. Otherwise, the Abel-Ruffini theorem informs us that no such general formula exists. $~$ Nevertheless, the rational root theorem or other polynomial factorization methods might prove helpful. However, I am afraid that recourse to numerical methods is ultimately unavoidable.
