In a terrible thunderstorm, along a rope ladder, \(n\) dwarfs ascend in a chain. If suddenly there is a thunderbolt, then from fear, every gnome, regardless of others, can fall with probability \(p\) (\(0 < p < 1\)). If the dwarf falls, then he also hits all of the dwarfs that are below him. Find:
a) The probability that exactly \(k\) dwarfs will fall.
b) The mathematical expectation of the number of fallen dwarfs.