Let S be the set of all passwords which are six to eight characters long, where each character is either an…

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 $={10}^8-5^8$

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

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

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

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