Exercise 2: A password is created using 8
distinct letters from the English alphabet (26
letters total). However, the password must
begin with a vowel (A, E, I, O, U), and the
remaining 7 letters can be any other distinct
consonants. How many distinct passwords can
be formed?

We are tasked with finding how many distinct passwords can be formed using 8 distinct letters, where the password must begin with a vowel, and the remaining 7 letters must be distinct consonants.

Let's break this down step-by-step:

### Step 1: Choose the first letter (vowel)
There are 5 vowels in the English alphabet: A, E, I, O, U. Since the first letter of the password must be a vowel, we have **5 choices** for the first letter.

### Step 2: Choose the remaining 7 letters (consonants)
After choosing a vowel, we are left with 21 consonants (since there are 26 letters in total, and 5 of them are vowels).

Now, we need to choose 7 distinct consonants from the 21 available consonants.

The number of ways to choose 7 distinct consonants from 21 is calculated using the permutation formula:
\[
P(n, r) = \frac{n!}{(n - r)!}
\]
where \(n\) is the total number of consonants (21) and \(r\) is the number of consonants we need to choose (7).

Thus, the number of ways to arrange the 7 consonants is:
\[
P(21, 7) = \frac{21!}{(21 - 7)!} = \frac{21!}{14!} = 21 \times 20 \times 19 \times 18 \times 17 \times 16 \times 15
\]

### Step 3: Calculate the total number of distinct passwords
The total number of distinct passwords is the product of the number of ways to choose the first vowel and the number of ways to arrange the remaining 7 consonants. Thus, the total number of passwords is:
\[
5 \times P(21, 7)
\]
\[
5 \times (21 \times 20 \times 19 \times 18 \times 17 \times 16 \times 15)
\]

Now, let's calculate this:
\[
5 \times (21 \times 20 \times 19 \times 18 \times 17 \times 16 \times 15) = 5 \times 2327925600 = 11639628000
\]

### Final Answer:
The total number of distinct passwords that can be formed is **11,639,628,000**.

---

Would you like more details on any part of the solution? Here are some related questions to explore:

1. How many distinct passwords can be formed if repetition of letters is allowed?
2. What if the password must start and end with a vowel?
3. How would the answer change if the password was 6 characters long instead of 8?
4. How many distinct passwords can be formed using exactly 2 vowels and 6 consonants?
5. What if the password was case-sensitive, considering both uppercase and lowercase letters?

**Tip:** When dealing with counting problems, it's useful to break them into distinct steps (like choosing a vowel first, then consonants) to avoid overcomplicating the process.

Exercise 2: A password is created using 8 distinct letters from the English alphabet (26 letters total). However, the password must begin with a vowel (A, E, I, O, U), and the remaining 7 letters can be any other distinct consonants. How many distinct passwords can be formed?

Explore how to calculate the number of distinct passwords that can be formed using 8 distinct letters, starting with a vowel and followed by 7 distinct consonants. This problem applies combinatorics and permutation concepts, suitable for students in Grades 9-12.

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