Previously, our password was 8 characters and included only lower-case letters. Now, because of changing requirements, we generate a random password with lower-case and upper-case letters, as well as numbers. How long of a password do we need in order to have the same security or better as before?

7 characters.

Explanation:

Here we have 26(small letters)+26(capital letters)+10(number from 0-9)=62

As we are having more number of possibilities for each character of the password, We will need a smaller length to reach the same or better level of security.

If we take 6 characters long password we will have 626 = 56,800,235,584‬ possibilities, which is less secure than the previous 8 characters long password.

But if we take 7 characters long password we will have 627 = 3,521,614,606,208 possibilities, which is almost 17 times larger than the previous password and has a smaller length when compared to the previous password.