Expected time of drawing all types of coins from a large pile

I've been working on the following question but am uncertain of how to solve it

Consider an infinitely large pile of coins. Each coin has a number {1, 2, . . . , n} written on it, and these numbers appear in equal propor- tions in the pile. Suppose that you keep drawing coins from this pile until the first time, T , that you have at least one coin of each number {1, 2, . . . , n}. Find the expected value of T.

Let $X_i(k):=\mathbb{P}\{ i^{th} \mbox{ type of coin is drawn for the first time on the }k^{th} \mbox{draw} \}$. This random variable follows a geometric distribution with parameter $p=\frac{1}{n}$.
Let $T:=\sum_{i=1}^n X_i$ then: \begin{equation} \mathbb{E}[T]=\sum_{i=1}^n\mathbb{E}[X_i]=\sum_{i=1}^n n = n^2. \end{equation}
• I don't think this is right unless the process the OP is thinking of requires you first to draw a specimen of coin $1$ before you start looking for coin $2$, draw a specimen of coin $2$ before you start looking for coin $3$, etc. – Brian Tung May 29 '15 at 16:59