Expected elevator stops for twelve riders

Twelve passengers board a lift on the ground level of a building with 10 upper floors. Each passenger independently picks one of the 10 floors uniformly at random as their destination. What is the expected number of floors where the lift stops for at least one passenger?

$10\left(1-0.9^{12}\right)\approx 7.18$. Write the stop count as a sum of 10 indicators, one per floor. A given floor is skipped only if all 12 riders choose another floor, probability $0.9^{12}$, so linearity of expectation gives $10(1-0.9^{12})$.