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 me this using set and standard mathematical symbols or format instead of explaining with words. 

To calculate the total number of valid passwords, let's use standard mathematical notation.

---

### Total possibilities for a password:
1. Passwords are 6 to 8 characters long.
2. Each character is an uppercase letter or a digit, so \( 26 + 10 = 36 \) possible characters per position.

Let \( L = 36 \) be the number of possible characters.

---

### Case 1: Password length \( n = 6 \)
#### Total passwords (with no restrictions):  
\[
L^6 = 36^6
\]

#### Passwords with no digits:  
If only uppercase letters are used (\(26\) choices):  
\[
26^6
\]

#### Valid passwords (at least one digit):  
\[
36^6 - 26^6
\]

---

### Case 2: Password length \( n = 7 \)
#### Total passwords:  
\[
L^7 = 36^7
\]

#### Passwords with no digits:  
\[
26^7
\]

#### Valid passwords:  
\[
36^7 - 26^7
\]

---

### Case 3: Password length \( n = 8 \)
#### Total passwords:  
\[
L^8 = 36^8
\]

#### Passwords with no digits:  
\[
26^8
\]

#### Valid passwords:  
\[
36^8 - 26^8
\]

---

### Total number of valid passwords:
Summing over all cases:
\[
\text{Total valid passwords} = (36^6 - 26^6) + (36^7 - 26^7) + (36^8 - 26^8)
\]

This expression provides the total number of possible passwords that meet the criteria.

Would you like me to compute this value numerically or provide further insights into this calculation

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?

Learn how to calculate the total number of valid passwords that are 6 to 8 characters long and include at least one digit. Explore combinatorics, exclusion principles, and detailed formulas to solve this problem step-by-step. Suitable for Grades 11-12.

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