Problem on waiting time until the $r$th success
Solution 1:
We have $$P(X=t+s)={{t-1+s}\choose{t-1}}/{{n}\choose{t}}$$ for $0\leq s\leq n-t$ by the stars and bars formula.
Calculating the EV will yield $$E(X)=\frac{(n+1)t}{t+1}.$$
Problem on waiting time until the $r$th success
Solution 1:
We have $$P(X=t+s)={{t-1+s}\choose{t-1}}/{{n}\choose{t}}$$ for $0\leq s\leq n-t$ by the stars and bars formula.
Calculating the EV will yield $$E(X)=\frac{(n+1)t}{t+1}.$$