A computer password is required to be 7 characters long. How many passwords are possible if the password requires 3 letter(s) followed by 4 digits (numbers 0-9), where no repetition of any letter or digit is allowed?

There are  possible passwords.

To solve the problem of determining how many possible passwords can be created under the given conditions, we will break it down step-by-step.

### Given:
- The password is 7 characters long.
- The first 3 characters must be letters, with no repetition.
- The next 4 characters must be digits (0-9), with no repetition.

### Step 1: Calculate the number of ways to choose the 3 letters
Since no repetition of letters is allowed, the total number of possible choices for the letters will decrease as we select each letter. There are 26 letters in the alphabet.

- The first letter has 26 options.
- The second letter has 25 options (since no repetition is allowed).
- The third letter has 24 options.

Thus, the number of ways to choose the 3 letters is:
\[
26 \times 25 \times 24 = 15,600
\]

### Step 2: Calculate the number of ways to choose the 4 digits
Similarly, since no repetition of digits is allowed and there are 10 digits (0-9):

- The first digit has 10 options.
- The second digit has 9 options.
- The third digit has 8 options.
- The fourth digit has 7 options.

Thus, the number of ways to choose the 4 digits is:
\[
10 \times 9 \times 8 \times 7 = 5,040
\]

### Step 3: Calculate the total number of possible passwords
Now, to find the total number of possible passwords, we multiply the number of ways to choose the letters by the number of ways to choose the digits:
\[
15,600 \times 5,040 = 78,624,000
\]

Thus, the total number of possible passwords is **78,624,000**.

Would you like more details, or do you have any questions?

Here are some related questions to consider:
1. How many passwords are possible if repetition of letters and digits is allowed?
2. How many passwords are possible if the password is required to be 8 characters long?
3. What if the password required 2 letters and 5 digits?
4. What if the digits could include the symbols "#" or "@"?
5. How many passwords are possible if only lowercase letters are allowed?

**Tip:** When no repetition is allowed, the number of options decreases with each successive choice.

A computer password is required to be 7 characters long. How many passwords are possible if the password requires 3 letter(s) followed by 4 digits (numbers 0-9), where no repetition of any letter or digit is allowed?

Learn how to calculate the total number of possible passwords for a 7-character password, requiring 3 letters followed by 4 digits with no repetition. This problem explores concepts in combinatorics, using the fundamental principle of counting. Suitable for students in Grades 8-10.

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