This is a problem from the second edition of Introduction to Probability by Blitzstein and Hwang.
Suppose there are $n$ types of toys, which you are collecting one by one. Each time you collect a toy, it is equally likely to be any of the $n$ types. What is the expected number of distinct toy types that you have after you have collected $t$ toys? (Assume that you will definitely collect $t$ toys, whether or not you obtain a complete set before then.)
I attempted to solve this by creating an indicator variable for each type of toy, $I_1,I_2,...,I_n$, each indicator equaling 1 if the toy is collected at least once in $t$ "collections" and 0 otherwise. The probability that a toy type is collected at least once in $t$ "collections" is $(1-(1-1/n)^t)$, and there are $n$ types of toys, so the total expected number of distinct toy types after collecting $t$ toys is $n(1-(1-1/n)^t)$. Is this correct?
