I recently encountered a problem involving the construction of equations involving factorials or combinatorial numbers. I recall reading somewhere (although I cannot recall the reference) that this can be a tricky subject.
In a particular scenario, the author of the problem suggested that the solution would be $x=1$. To construct the simplest equation possible with this solution, the author proposed using $x-1=0$. Subsequently, the author manipulated this equation to obtain:
\begin{align} (x-1)^2 &= 0 \\ x^2 - 2x + 1 &= 0 \\ x(x-2) + 1 &= 0 \\ x(x-2) &= -1 \\ \frac{x!}{(x-1)!}\frac{(x-2)!}{(x-3)!} &= -1 \\ x!(x-2)! &= -(x-1)!(x-3)! \end{align}
However, upon substituting $x=1$ into the equations, we encounter expressions such as $(-1)!$ and $(-2)!$, which do not make sense in this context.
My question is: Am I missing something, or is the author's approach incorrect? Any insights or explanations would be greatly appreciated. Thank you very much.
The factorial can be generalized to complex arguments using the gamma function: $n!=\Gamma(n+1)$. This function has poles at all non-positive integers, so the factorial of a negative integer diverges. Thus you’re right to think that this attempt at constructing an equation with solution $x=1$ is misguided, since substituting $x=1$ leads to expressions that are divergent even when appropriately generalized.