Find the roots of the equation $1-x+\frac{x(x-1)}{2!}+\frac{x(x-1)(x-2)}{3!}+\ldots+(-1)^{n} \frac{x(x-1) \ldots(x-n+1)}{n!}=0$.
The equality $1-k+\frac{k(k-1)}{2}+\ldots+(-1)^{k} \frac{k(k-1) \ldots \cdot 2 \cdot 1}{k!}=(1-1)^{k}=0$ shows that the numbers $k=$ $1,2, \ldots, n$ are roots of this equation. The equation cannot have more than $n$ roots, since its degree is $n$.

## Answer

$1,2, \ldots, n$