The total number of 5-digit numbers of different digits in which the digit in the middle is the largest is

The smallest number, which can occur in the middle is 4. The number of numbers with 4 in the middle${=}^4{\mathrm{P}}_4{-}^3{\mathrm{P}}_3$.

(∴ The other four places are to be filled by 0, 1, 2 and 3, and a number can not begin with 0)

Similarly, the number of numbers with 5 in the middle = ${ }^5{\mathrm{P}}_4{-}^4{\mathrm{P}}_3$, etc.

∴ The required number of numbers

$\begin{array}{l}=\left({ }^4{\mathrm{P}}_4{-}^3{\mathrm{P}}_3\right)+\left({ }^5{\mathrm{P}}_4{-}^4{\mathrm{P}}_3\right)+\left({ }^6{\mathrm{P}}_4{-}^5{\mathrm{P}}_3\right)+\dots +\left({ }^9{\mathrm{P}}_4{-}^8{\mathrm{P}}_3\right)\\ =\sum _{\mathrm{n}=4}^9 \left({ }^{\mathrm{n}}{\mathrm{P}}_4{-}^{\mathrm{n}-1}{\mathrm{P}}_3\right)\end{array}$