Final answer:
The number of possible passwords is calculated by taking 62 to the power of each possible password length (8 to 12) and summing them. This concept relies on the permutations principle in mathematics.
Step-by-step explanation:
Calculating the number of possible passwords that can be created with 8 to 12 characters using lowercase, uppercase, and numbers involves understanding permutations. Each position in the password can contain a character from a set of 62 possible characters (26 lowercase + 26 uppercase + 10 numbers). For an 8-character password, there are 62^8 possibilities, for 9 characters 62^9, and so on, up to 12 characters, which would be 62^12 possibilities.
To find the total number of possible passwords, we need to sum these individual totals:
Total number of passwords = 62^8 + 62^9 + 62^10 + 62^11 + 62^12
This is a large computational problem that is best solved using a calculator or computer software. However, understanding the concept of raising the number of possible characters to the power of the length of the password is a critical step in comprehending how permutations work.