A carnival sharpshooter game charges $\$25$ for $25$ shots at a target.
If the shooter hits the bullseye fewer than $5$ times he gets no prize.
If the shooter hits the bullseye $5$ times he gets back $\$10$.
For each additional bullseye over $5$ he gets back an additional $\$5$.
The shooter estimates that he has a $0.2$ probability of hitting the bullseye on any given shot.
What is the shooter's expected gain if he plays the game?
I know I can calculate this by brute force, but is there a faster way to solve this than multiplying 0-25 by their respective rewards?