The expectation of total number of different states in N time points [Conditions] 
(1) An object has K possible states. 
(2) This object can have only one state at a single time point. 
(3) The probability of each state at any single time point is 1/K, and each time point is independent of another. 
[Question] What is the expectation of different states appearing in N time points? (Obviously, this expectation should be less than K. )
Someone told me the solution is $K(1-(1-\frac{1}{K})^N)$, but I could not understand his explanation. 
So here I hope you to validate this solution, and provide detailed explanation. 
 A: To give the answer you quote, you need a stronger version of (3) so that the states at each time point are independent of each other.
Hints:


*

*What is the probability that at any particular time point the object has a partiuclar state $x$?

*What is the probability that at any particular time point the object does not have state $x$?

*What is the probability that at all $N$ time points the object does not have state $x$?

*What is the probability that at at least one of $N$ time points that the object has state $x$?

*What is the expectation of an indicator that at at least one of $N$ time points the object has state $x$?

*What is the sum over all $K$ states of the expectations of an indicator that at at least one of $N$ time points the object has each state? (Use linearity of expectation)

*What is the expectation of the sum over all $K$ states of an indicator that at at least one of $N$ time points the object has each state?

*What is the expectation of the number of states the object has over $N$ time points?

