HMMT 2025 U24

For any integer $x$ , let

$$f(x) = 100! \left(1 + x + \frac{x^{2}}{2!} + \frac{x^{3}}{3!} + \dots + \frac{x^{100}}{100!}\right).$$

A positive integer $a$ is chosen such that $f(a) - 20$ is divisible by $101^{2}$ . Compute the remainder when $f(a + 101)$ is divided by $101^{2}$ .