The probability that any one person gets off on floor i is 1/10, and the probability that he does not get off is 9/10. So the probability that no one gets off on floor i is . Then
The expected value of the number of floors where the elevator stops is then
Here we have made use of the theorem that E(X+Y) = E(X) + E(Y). It's important to realize that the theorem does not require that X and Y be independent. That's good for us in this problem, because the 's are not independent.