Total available characters

Each character can be chosen from:

• alphabets: $\{A,B,C,D,E\}$ → $5$ choices
• numbers: $\{1,2,3,4,5\}$ → $5$ choices

So total choices for each position = $10$.

Possible password lengths

Passwords are of length $6,7,$ or $8$.

Count passwords with at least one number

We use complement counting.

For a fixed length $n$:

• total passwords = $10^n$
• passwords with only alphabets = $5^n$

Therefore, passwords with at least one number = $10^n - 5^n$

Sum over all allowed lengths

Hence required number is $\sum_{n=6}^{8} (10^n - 5^n)$

Compute: $= (10^6-5^6) + (10^7-5^7) + (10^8-5^8)$

Factor out $5^6$

Write each term in powers of $5$: $10^6 = (2\cdot 5)^6 = 2^6 5^6 = 64\cdot 5^6$ $10^7 = 2^7 5^7 = 128\cdot 5\cdot 5^6 = 640\cdot 5^6$ $10^8 = 2^8 5^8 = 256\cdot 25\cdot 5^6 = 6400\cdot 5^6$ Also, $5^7 = 5\cdot 5^6, \qquad 5^8 = 25\cdot 5^6$

So, $10^6-5^6 = (64-1)5^6 = 63\cdot 5^6$ $10^7-5^7 = (640-5)5^6 = 635\cdot 5^6$ $10^8-5^8 = (6400-25)5^6 = 6375\cdot 5^6$

Add coefficients

Therefore, $\alpha = 63+635+6375 = 7073$

Final answer

$\boxed{7073}$

Comparison with stored answer

Stored correct answer = $7073$.

This matches our derived answer.