Upper A 9​-character password computer must satify the following​ conditions:
2 letters​ (case is not​ sensitive), chosen from the 26 letters of the alphabet
4 ​digits, chosen from the set​ {0, 1,​ 2, 3,​ 4, 5,​ 6, 7,​ 8, 9}
3 ​symbols, chosen from the set​ {#, @,​ $, %}
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Part 1
a.  How many passwords can be created if the 9 characters are chosen without replacement​?
select: 28,478,668,800,000
enter your response here ways
b.  How many passwords can be created if the 9 characters are chosen with replacement​?
select: 156,043,161,600,000
enter your response here ways

To solve this problem, we'll compute the number of possible passwords for both cases: without replacement and with replacement.

### Part a: Without Replacement
Since the characters are chosen **without replacement**, each chosen character cannot be reused.

1. **2 letters**: There are 26 letters in the alphabet, and since case is not sensitive, the number of ways to choose 2 letters is:
   \[
   \text{Ways to choose letters} = 26 \times 25 = 650
   \]
   (because once one letter is chosen, the next one cannot be the same).

2. **4 digits**: There are 10 digits (from 0 to 9). Since digits cannot be reused, the number of ways to choose 4 digits is:
   \[
   \text{Ways to choose digits} = 10 \times 9 \times 8 \times 7 = 5040
   \]

3. **3 symbols**: There are 4 symbols (#, @, $, %). Since symbols cannot be reused, the number of ways to choose 3 symbols is:
   \[
   \text{Ways to choose symbols} = 4 \times 3 \times 2 = 24
   \]

Therefore, the total number of passwords that can be created without replacement is:
\[
\text{Total without replacement} = 650 \times 5040 \times 24 = 78,624,000
\]

---

### Part b: With Replacement
Now, since the characters are chosen **with replacement**, each character can be reused.

1. **2 letters**: There are 26 letters, and since they can be reused, the number of ways to choose 2 letters is:
   \[
   \text{Ways to choose letters} = 26 \times 26 = 676
   \]

2. **4 digits**: Since digits can be reused, the number of ways to choose 4 digits is:
   \[
   \text{Ways to choose digits} = 10 \times 10 \times 10 \times 10 = 10,000
   \]

3. **3 symbols**: Since symbols can be reused, the number of ways to choose 3 symbols is:
   \[
   \text{Ways to choose symbols} = 4 \times 4 \times 4 = 64
   \]

Therefore, the total number of passwords that can be created with replacement is:
\[
\text{Total with replacement} = 676 \times 10,000 \times 64 = 432,640,000
\]

### Final Answers:
- **Part a**: 78,624,000 passwords can be created without replacement.
- **Part b**: 432,640,000 passwords can be created with replacement.

Let me know if you want more details or have any questions.

Here are 5 questions that expand this information:
1. How does the concept of "without replacement" impact the calculation for each type of character?
2. What happens if the case is sensitive for the letters? How would the calculations change?
3. How would the password complexity change if we allow special symbols or digits to repeat more than once?
4. How would the total number of combinations change if we allow mixed case letters (upper and lower case)?
5. Can you calculate the probability of guessing one specific password out of all possible combinations?

**Tip**: Password strength increases exponentially with length and diversity of characters used (letters, numbers, symbols). Always choose a longer, more complex password for better security.

Upper A 9​-character password computer must satisfy the following​ conditions: 2 letters​ (case is not​ sensitive), chosen from the 26 letters of the alphabet; 4 ​digits, chosen from the set​ {0, 1,​ 2, 3,​ 4, 5,​ 6, 7,​ 8, 9}; 3 ​symbols, chosen from the set​ {#, @,​ $, %}. Part 1: a. How many passwords can be created if the 9 characters are chosen without replacement? b. How many passwords can be created if the 9 characters are chosen with replacement?

Explore how to calculate the number of 9-character passwords that can be created using a combination of letters, digits, and symbols. Learn the formulas for counting with and without replacement, covering concepts like combinatorics and the multiplication principle. Suitable for high school students.

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