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 calculate the total number of possible passwords, we need to multiply the number of choices for each section of the password.

1. **Letters (3 letters):** There are 26 letters in the alphabet, and since letters can be repeated, each of the 3 letters has 26 possible choices. So, the total number of ways to choose the 3 letters is:

   \[
   26 \times 26 \times 26 = 26^3
   \]

2. **Symbols (2 symbols):** The company allows the use of 11 different symbols, and symbols can also be repeated. So, the total number of ways to choose the 2 symbols is:

   \[
   11 \times 11 = 11^2
   \]

3. **Digits (4 digits):** There are 10 digits (0 through 9), and digits can be repeated. So, the total number of ways to choose the 4 digits is:

   \[
   10 \times 10 \times 10 \times 10 = 10^4
   \]

Finally, to find the total number of possible passwords, multiply the results from each part:

\[
26^3 \times 11^2 \times 10^4
\]

Now, calculating:

\[
26^3 = 26 \times 26 \times 26 = 17,576
\]
\[
11^2 = 11 \times 11 = 121
\]
\[
10^4 = 10,000
\]

So, the total number of possible passwords is:

\[
17,576 \times 121 \times 10,000 = 2,125,696,000
\]

Thus, there are **2,125,696,000** possible passwords.

---

Would you like a breakdown of any specific part or further details? Here are some related questions you could explore:

1. What if symbols couldn't be repeated?
2. What if letters couldn't be repeated?
3. How would the number of passwords change if the number of digits were reduced to 3?
4. What is the probability of guessing a password correctly on the first try?
5. How does the length of a password affect security?

**Tip:** Increasing the number of unique symbols, letters, or digits significantly boosts the total number of possible combinations, enhancing password security.

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?

Learn how to calculate the total number of possible password combinations consisting of 3 letters, 2 symbols, and 4 digits. This detailed breakdown uses the multiplication principle of counting, exploring choices for letters, symbols, and digits. Suitable for high school students and beyond.

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