Book problem: What is the probability $P[k]$ of a run of length $k$ for rolling a die until a duplicate number appears?
My answer: $P[k] = 1 - 1(\frac{5}{6})(\frac{4}{6})\cdots(\frac{7-k}{6})$ for $k$ between $1$ and $6$; $P[k] = 1$ otherwise.
Book answer: $1 - 1(1 - \frac{1}{6})(1 - \frac{2}{6})\cdots(1 - \frac{k}{6})$.
Question: Is there a typo in their answer? When $k = 1$ I'd expect a probability of $0$, for example.
My own reasoning went as follows.
For a run of length $3$, so $k = 3$, to get no duplicates the probability would be $\frac{6\cdot 5\cdot 4}{6^3}$ so that to get at least one pair of duplicates we'd take the complement to get the answer.
Book: "Methods of Mathematics" by Hamming