P1(a) Question:
P1: (a) Each user in a computer system has a password which is 6 to 8 characters long where each character is an uppercase letter or a digit. Each password must contain at least one digit. How many possible passwords are there?

Calculate this problem using set.

To calculate the number of possible passwords that meet the given criteria using a **set-based approach**, follow these steps:

### Step 1: Define the set of possible characters.
The characters available are **uppercase letters** and **digits**:
- Uppercase letters: 26 characters (A-Z)
- Digits: 10 characters (0-9)

Thus, the total number of available characters is:
\[
26 + 10 = 36
\]

### Step 2: Total passwords (no restrictions).
The total number of passwords of length $ n $ (6 to 8 characters) is:
\[
36^n
\]
This represents all possible combinations of characters for a given password length.

For lengths of 6, 7, and 8, the total is:
\[
\text{Total passwords} = 36^6 + 36^7 + 36^8
\]

### Step 3: Subtract passwords with no digits.
Passwords with **no digits** consist only of uppercase letters (26 choices per character). The total number of such passwords for a given length $ n $ is:
\[
26^n
\]

For lengths of 6, 7, and 8, the total is:
\[
\text{Passwords with no digits} = 26^6 + 26^7 + 26^8
\]

### Step 4: Subtract the invalid passwords.
The valid passwords are those that contain **at least one digit**. Using the complement principle:
\[
\text{Valid passwords} = \text{Total passwords} - \text{Passwords with no digits}
\]

Substituting the values:
\[
\text{Valid passwords} = (36^6 + 36^7 + 36^8) - (26^6 + 26^7 + 26^8)
\]

### Step 5: Perform the calculations.
Let's compute each term:
1. $ 36^6 = 2,176,782,336 $
2. $ 36^7 = 78,364,164,096 $
3. $ 36^8 = 2,821,109,907,456 $

Thus:
\[
36^6 + 36^7 + 36^8 = 2,176,782,336 + 78,364,164,096 + 2,821,109,907,456 = 2,901,650,853,888
\]

Similarly:
1. $ 26^6 = 308,915,776 $
2. $ 26^7 = 8,031,810,176 $
3. $ 26^8 = 208,827,064,576 $

Thus:
\[
26^6 + 26^7 + 26^8 = 308,915,776 + 8,031,810,176 + 208,827,064,576 = 217,167,790,528
\]

Finally, subtract:
\[
\text{Valid passwords} = 2,901,650,853,888 - 217,167,790,528 = 2,684,483,063,360
\]

### Final Answer:
The total number of valid passwords is:
\[
\boxed{2,684,483,063,360}
\]

---

Would you like a detailed breakdown of any specific step? Here are related questions for exploration:

1. What would the result be if the password length were fixed at 7 characters?
2. How does the inclusion of lowercase letters affect the calculation?
3. What is the probability that a random password contains exactly two digits?
4. How would the calculation change if special characters (e.g., `@, #, $`) were included?
5. What is the average number of digits in a valid password?

**Tip**: Always use the complement principle for "at least one" constraints to simplify combinatorial

Each user in a computer system has a password which is 6 to 8 characters long where each character is an uppercase letter or a digit. Each password must contain at least one digit. How many possible passwords are there?

Calculate the total number of valid passwords of 6 to 8 characters, where each character is an uppercase letter or digit, and each password must contain at least one digit. This combinatorial problem is solved using counting principles and the complement principle, resulting in a total of 2,684,483,063,360 valid passwords.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation