Let's count the number of passwords of size $k$ that consist a combination of lowercase letters and digits with at least one digit.
We know that for each char in the password we have $10 + 26 =36$ possible chars ($10$ different digits, and $26$ different lower letters). So the total passwords of size $k$ are $36^k$ - for each char out of $k$ chars we have $36$ different possibilities. But not all passwords consist at least one digit, we'll count those - each char in the password must be a lowercase letter- so there are $26^k$ such passwords.
Combining those results we get that the number of passwords of size $k$ that have at least one digit is $36^k - 26^k$.
In, our case, we just need to sum the disjoint cases of passwords of size $6,7,8$ and get that the number of such passwords is $$ 36^6 - 26^6 + 36^7 - 26^7 + 36^7 - 26^7 $$