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Combinatorics is the branch of mathematics that focuses on counting, arrangement, and combination of objects. For password security, it helps us calculate the total possible number of passwords based on given criteria. Let’s break this down:
Consider a password composed of lowercase and uppercase letters, digits, and special characters all combined. For each character in the password, there are 68 choices (26 lowercase letters + 26 uppercase letters + 10 digits + 6 special characters).
If a password has a length of 'n' characters, the number of possible passwords can be calculated by raising the number of choices to the power of 'n'. For example, if 'n' is 8, then the number of passwords of length 8 is given by \(68^8\).
This formula extends to passwords of different lengths as well, such as 9, 10, 11, and 12 characters, and the total number of possible passwords is the sum of these values:
\(68^8 + 68^9 + 68^{10} + 68^{11} + 68^{12}\). So, combinatorics helps us understand the vast number of combinations even a seemingly short password can generate.

Password strength is crucial for securing systems. It refers to how difficult it is for unauthorized users (hackers) to guess or crack a password. Various factors, such as length and character variety, affect password strength.
The more combinations a password has, the stronger it is. For instance, a password of 12 characters drawn from 68 possible characters is stronger than an 8-character password.
To ensure a robust password, including special characters significantly increases the number of possible combinations. We calculated the total number of passwords containing at least one special character by subtracting the number of passwords without special characters from the total number of passwords. This operation emphasizes the complexity added by simple variations, which significantly enhances password strength.

Time complexity in password security relates to the amount of time a hacker would need to exhaust all possible password combinations. When we say 'time complexity,' we often refer to how this scales with the length and complexity of the password.
Let's take the scenario where a hacker tries every possible password and it takes one nanosecond per attempt. For 8 to 12 characters, that’s a total of approximately \(1.29 \times 10^{22}\) passwords.
Multiplying that by the time per attempt (1 nanosecond), we get \(1.29 \times 10^{22}\) nanoseconds. To convert this enormous number into years, divide by the number of nanoseconds in a year (\(3.15 \times 10^{16}\)). This equates to roughly 409,524 years.
This calculation shows just how impractical it is for a hacker to break such a highly complex password, emphasizing the essence of choosing long and varied passwords.