A company has its employees choose a password which consists of 9 characters: 3 letters, followed by 2 symbols, followed by 4 digits. There are a total of 11 symbols that the employees can choose from. Letters, symbols and digits can be repeated. How many possible passwords are there?

To find the total number of possible passwords, we will calculate the number of choices for each component of the password (letters, symbols, and digits), and then multiply these together, since each component is chosen independently.

### Password structure:
The password consists of:
- 3 letters
- 2 symbols
- 4 digits

#### Step 1: Number of choices for the letters
The total number of letters in the English alphabet is 26. Since letters can be repeated, for each of the 3 positions we have 26 choices.

Thus, the total number of ways to choose 3 letters is:
\[
26 \times 26 \times 26 = 26^3 = 17,576
\]

#### Step 2: Number of choices for the symbols
There are 11 symbols to choose from, and since symbols can be repeated, for each of the 2 positions we have 11 choices.

Thus, the total number of ways to choose 2 symbols is:
\[
11 \times 11 = 11^2 = 121
\]

#### Step 3: Number of choices for the digits
There are 10 digits (0 through 9), and since digits can be repeated, for each of the 4 positions we have 10 choices.

Thus, the total number of ways to choose 4 digits is:
\[
10 \times 10 \times 10 \times 10 = 10^4 = 10,000
\]

#### Step 4: Total number of possible passwords
Now, multiply the number of choices for each part together:
\[
26^3 \times 11^2 \times 10^4 = 17,576 \times 121 \times 10,000
\]

Calculating:
\[
17,576 \times 121 = 2,127,696
\]
\[
2,127,696 \times 10,000 = 21,276,960,000
\]

Thus, the total number of possible passwords is:
\[
\boxed{21,276,960,000}
\]

This is the total number of distinct passwords the employees can create under these conditions.

Would you like more details or have any questions?

### Related Questions:
1. What if the symbols couldn't be repeated? How would that change the answer?
2. What if the letters couldn't be repeated? How would that affect the result?
3. How many possible passwords are there if the structure of the password is reversed (digits, symbols, letters)?
4. What if the password had 4 letters, 3 symbols, and 2 digits instead?
5. How does the total number of possible passwords change if the company adds 2 more symbols to the list?

### Tip:
When calculating combinations or permutations, remember that if repetition is allowed, the number of options remains the same for each position; if not, the number of options decreases for each subsequent choice.

Discover how to calculate the number of possible passwords when repetition is allowed in letters, symbols, and digits. This problem involves key combinatorial concepts, such as the fundamental counting principle, and is suitable for high school students in Grades 10-12.

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