How many integers between $1000$ and $9999$ have distinct digits?

Question

I computed the answer to be $10 \cdot 9 \cdot 8 \cdot 7$, but apparently we had to do $9 \cdot 9 \cdot 8 \cdot 7$.

But there are $10$ digits between $0$ and $9$, right? Can someone explain to me how this is possible?

Answer 1

Hint: The first of your four digits can't be $0$ if the number is between $1000$ and $9999$.