You need to have a password with 6  letters followed by 3 odd digits between 0 and 9, inclusive.  If the characters and digits cannot be used more than once, how many choices do you have for your password?



To determine the number of possible passwords with 6 letters followed by 3 odd digits, we need to break down the problem into two parts: selecting the letters and selecting the digits. Here’s the step-by-step process:

### 1. Selecting the letters:
- The password requires 6 letters, and since letters cannot be repeated, we are choosing from a set of 26 unique letters in the English alphabet.
- For the first letter, we have 26 choices.
- For the second letter, we now have only 25 choices (since one letter has already been used).
- For the third letter, we have 24 choices, and so on, down to 21 choices for the sixth letter.

The number of ways to select the 6 letters is:
\[
26 \times 25 \times 24 \times 23 \times 22 \times 21
\]

### 2. Selecting the odd digits:
- The password requires 3 odd digits, and the digits cannot be repeated.
- The odd digits between 0 and 9 are: 1, 3, 5, 7, and 9, giving us 5 possible choices for the first digit.
- After selecting the first odd digit, we have 4 remaining choices for the second digit.
- After selecting the second odd digit, we have 3 remaining choices for the third digit.

The number of ways to select the 3 odd digits is:
\[
5 \times 4 \times 3
\]

### 3. Total number of possible passwords:
Now, to find the total number of possible passwords, we multiply the number of ways to select the letters by the number of ways to select the digits:
\[
(26 \times 25 \times 24 \times 23 \times 22 \times 21) \times (5 \times 4 \times 3)
\]

Let’s calculate that:

\[
26 \times 25 \times 24 \times 23 \times 22 \times 21 = 165,765,600
\]
\[
5 \times 4 \times 3 = 60
\]
\[
165,765,600 \times 60 = 9,945,936,000
\]

Thus, the total number of choices for the password is **9,945,936,000**.

---

Would you like any further details or clarifications?

Here are 5 related questions to think about:
1. How would the calculation change if the letters or digits could be repeated?
2. What if the number of letters in the password was increased to 7?
3. How many passwords could be formed if the digits were not restricted to odd numbers?
4. What if the letters had to be all uppercase or lowercase?
5. How many passwords could be formed if one special character was added?

**Tip:** When dealing with large numbers of combinations, breaking the problem into smaller steps makes it easier to manage and compute.

You need to have a password with 6 letters followed by 3 odd digits between 0 and 9, inclusive. If the characters and digits cannot be used more than once, how many choices do you have for your password?

Learn how to calculate the number of possible passwords with 6 unique letters followed by 3 unique odd digits from 0-9. This problem uses the principle of counting and permutation formulas to break down the steps. Suitable for high school students in Grades 9-12.

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