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

Finding the number of positive integers between $1000$ and $9999$ inclusive that have distinct digits:

• The positive integers between $1000$ and $9999$ inclusive have $4$ digits.
• The first digit cannot be zero because otherwise, it will become a $3$ digit integer. So, the first digit can be filled in $9$ different ways.
• The second digit can be filled by $9$ different ways because the second digit cannot be the same as that of the first digit but zero can be the second digit.
• The third digit can be filled by $8$ different ways because the third digit cannot be the same as that of the first and second digits.
• The fourth digit can be filled in $7$ different ways because the fourth digit cannot be the same as that of the first, second, and third digit.

Hence, by using the product rule the number of positive integers between $1000$ and $9999$ inclusive that have distinct digits can be calculated as:

$\begin{array}{rcl}\mathrm{number}\mathrm{of}\mathrm{positive}\mathrm{integers}& =& 9\times 9\times 8\times 7\\ & =& 4536\end{array}$

Hence, $4536$ positive integers between $1000$ and $9999$ inclusive have distinct digits.