HMMT November 2024 U36

Estimate the value of

$$\frac{20!\cdot 40!\cdot 40!}{100!} \cdot \sum_{i=0}^{40} \sum_{j=0}^{40} \frac{(i+j+18)!}{i!j!18!}$$

Submit a positive real number $E$ either in decimal or in a fraction of two positive integers written in decimal (such as $\frac{2024}{2025}$ ). If the correct answer is $A$, your will receive $\max (20-\lfloor 5000 \cdot|E-A|\rfloor, 0)$ points.