A password requires exactly 6 characters and no character may show up more than once. The first 3 characters must contain digits (0-9). The next 3 charecters must contain letters only. The letters are case sensitive, so they may be upper or lower case. How many possible combinations exist?

To determine how many possible combinations of such passwords exist, let's break down the problem step by step.

### Step 1: First 3 characters (digits)
The first 3 characters of the password must be digits (0-9), and no digit can repeat. There are 10 possible digits (0, 1, 2, ..., 9). Since no digit can be used more than once, the number of ways to choose the first 3 digits is:

- For the first digit, there are 10 possible choices.
- For the second digit, there are 9 remaining choices.
- For the third digit, there are 8 remaining choices.

Thus, the total number of ways to choose the first 3 characters (digits) is:
\[
10 \times 9 \times 8 = 720
\]

### Step 2: Last 3 characters (letters)
The last 3 characters of the password must be letters, and they are case-sensitive. There are 26 letters in the alphabet, and since both uppercase and lowercase are allowed, there are \(26 \times 2 = 52\) possible choices for each letter. Again, no letter may repeat, so the number of ways to choose the last 3 characters (letters) is:

- For the first letter, there are 52 possible choices.
- For the second letter, there are 51 remaining choices.
- For the third letter, there are 50 remaining choices.

Thus, the total number of ways to choose the last 3 characters (letters) is:
\[
52 \times 51 \times 50 = 132,600
\]

### Step 3: Total number of possible combinations
To find the total number of possible password combinations, we multiply the results from Step 1 and Step 2:

\[
720 \times 132,600 = 95,472,000
\]

### Conclusion:
The total number of possible password combinations is **95,472,000**.

---

Would you like more details or have any questions? Here are 5 related questions:

1. How would the answer change if the letters were not case-sensitive?
2. What if repetition of characters was allowed?
3. How many passwords would exist if the password required exactly 8 characters with the same rules?
4. How many combinations would there be if the password was required to start with a letter?
5. What if the password required only 4 characters instead of 6?

**Tip:** When dealing with combinations that have restrictions (like no repetition), always calculate the choices step by step, accounting for the reduced number of options as you progress.

A password requires exactly 6 characters and no character may show up more than once. The first 3 characters must contain digits (0-9). The next 3 characters must contain letters only. The letters are case sensitive, so they may be upper or lower case. How many possible combinations exist?

Explore the calculation of possible combinations for a 6-character password with 3 unique digits and 3 unique case-sensitive letters. Learn step-by-step how to apply combinatorics and permutations to find the total number of possible passwords, 95,472,000.

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