Question

There is a three-digit password and it is known that each digit can have four values \(5,6,7\) or 8 . If there is exactly one correct password, how many distinct wrong passwords are there? (1) 63 (2) 80 (3) 81 (4) 64

Short answer

Answer: 63.

Step-by-step solution

Find Total Possible Passwords

We have 4 choices for each of the 3 digits, leading us to have a total of \(4 \times 4 \times 4\) possible passwords. So, the total possible passwords are \(4 \times 4 \times 4 = 64\).

Subtract 1 Correct Password from Total

Since there is exactly one correct password among these total possible passwords, we need to subtract that from our total to find the number of distinct wrong passwords. Therefore, the number of distinct wrong passwords is \(64 - 1 = 63\). The answer is (1) 63.

Key concepts

Permutations

The concept of permutations is crucial to understanding many combinatorics problems, including those involving password combinations. A permutation is an arrangement or sequence of items where the order matters. For instance, when creating a password, the sequence in which letters or numbers appear is essential, as ‘123’ is not the same as ‘321’. In mathematical terms, to determine the number of permutations of a set with ‘n’ unique items, we use the factorial function represented as ‘n!’, which is the product of all positive integers less than or equal to ‘n’.

In our three-digit password example, each digit can be one of four values (5, 6, 7, or 8). The permutations here are not factorial because each digit is independent and can be chosen repeatedly. This means that for a three-digit password, we multiply the number of options for each digit's placement. The calculation is then as simple as elevating the number of choices to the power of the password length: in this case, it's four choices to the third power, yielding a total of 64 possible permutations.

Password Combinations

Password combinations often apply the concept of permutations, but with a specific focus on creating secure access codes or locks.

Creating a password involves selecting a series of characters where various rules might apply, such as no repeating characters, or characters must be from different character sets (like uppercase letters, lowercase letters, numbers, and symbols). The example of a three-digit password where each digit can be 5, 6, 7, or 8 is a simple case. However, secure passwords typically require more complexity, including a mix of character types and longer lengths.

The number of wrong passwords, or more technically 'incorrect combinations,' is essentially the total permutations minus any restrictions. In our textbook problem, there is exactly one correct password, so we start with all the possible combinations (64) and subtract the one correct combination to arrive at 63 incorrect ones. Understanding this principle is crucial for setting up strong passwords in real-world applications, where limiting the number of simple or common passwords increases security.

Mathematical Problem-Solving

Mathematical problem-solving is a systematic approach that involves understanding the problem, devising a plan, carrying out the plan, and looking back to check the results. In our password problem, we used basic combinatorics to devise our plan.

Step 1 involved understanding how many total possible passwords there are, based on the given parameters. In this case, the parameters were a three-digit password and four possible values for each digit. Step 2 of the plan required us to consider any constraints given in the problem; here, it was the fact that there is one correct password. After carrying out these steps, calculating the number of permutations and then subtracting the one correct one, we arrived at the conclusion that there are 63 incorrect passwords. This systematic process can be applied to various mathematical problems and is a key skill in problem-solving.

Improving Comprehension:

To help students better grasp this process, it’s beneficial to practice with different problem types, review permutation and combination formulas, and understand each problem’s unique constraints. Applying this strategy helps in transforming complex problems into a series of manageable steps.