Let S be the set of all passwords which are six to eight characters...

Given total available characters is $10$ and total digits are $5$,

So, for$6 -$digit password, the number of ways $= {10}^{6} - {5}^{6}$

For $7 -$digit password, the number of ways $= {10}^{7} - {5}^{7}$

For $8 -$digit password, the number of ways $= {10}^{8} - {5}^{8}$

$∴$ Total ways $= \left({10}^{6} + {10}^{7} + {10}^{8}\right) - \left({5}^{6} + {5}^{7} + {5}^{8}\right)$

$= {2}^{6}{5}^{6} + {2}^{7}{5}^{7} + {2}^{8}{5}^{8} - {5}^{6} - {5}^{7} - {5}^{8}$

$= {5}^{6}\left({2}^{6} + 5 \cdot {2}^{7} + 25 \cdot {2}^{8} - 1 - 5 - 25\right) = 7073 \cdot {5}^{6}$

Now on comparing with $\alpha \times {5}^{6}$ we get $\alpha = 7073$