Seven people enter a lift.the lift stops at three unspecified floors.at each of the three floors,no one enters the lift,but at least one person leaves the lift.after the three floor stops the lift is empty.in how many ways can this happen?
I tried to use the principle of inclusion and exclusion. I took P1 to be the property in which Ist person was left in the lift and P2 to be the property in which 2nd person was left in the lift and so on.Note P5,P6 and P7 are not possible. Therefore $n(1)=3^67$ $n(2)=C^7_23^5$ $n(3)=C^7_33^4$ $n(4)=C^7_43^3$
So number of ways=$$3^7-n(1)+n(2)-n(3)+n(4)$$ But that gave me a wrong answer. I couldn't understand where I went wrong.please help me. Thanks.