From the image, the question involves computing the number of passwords (or strings) satisfying specific constraints. Here's a detailed breakdown of the problem and solution:

Definitions from the image:

1. Digits: {0, 1, 2, 3, 4, 5, 6, 7, 8, 9} (10 elements)
2. Letters: {a, b, c, ..., z} (26 elements)
3. Special Characters: {*, &, $, #} (4 elements)

Constraints to solve:

1. Part (i): Strings of length 7. Characters can be special characters, digits, or letters, with no repeated characters.
2. Part (ii): Strings of length 6. Characters can be special characters, digits, or letters, with no repeated characters, and the first character cannot be a special character.

Solution:

Step 1: Total number of available characters

• Total = 10 (digits) + 26 (letters) + 4 (special characters) = 40 characters.

Part (i): Length-7 strings with no repeated characters

• Since no repeated characters are allowed:
  • For the first position: Any of the 40 characters.
  • For the second position: 39 remaining characters.
  • For the third position: 38 remaining characters.
  • Continue this pattern...

Thus, the total number of strings is: $40 \times 39 \times 38 \times 37 \times 36 \times 35 \times 34$

Part (ii): Length-6 strings with first character not being a special character

• Available first character choices: Only digits or letters (10 + 26 = 36 options).
• Remaining characters: Any of the remaining 39 characters (after the first is chosen), with no repeats.

For the total number of strings: $36 \times 39 \times 38 \times 37 \times 36 \times 35$

Calculations:

I'll compute these values for precise results. One moment!# Defining the total number of available characters in each category total_characters = 40 non_special_first = 36

Calculating Part (i): 40 x 39 x 38 x 37 x 36 x 35 x 34

part_i = 1 for i in range(40, 33, -1): # From 40 to (40 - 7 + 1) part_i *= i

Calculating Part (ii): 36 x 39 x 38 x 37 x 36 x 35

part_ii = non_special_first for i in range(39, 34, -1): # From 39 to (39 - 6 + 1) part_ii *= i

part_i, part_ii(93963542400, 2487270240)(93963542400, 2487270240)success### Results:

1. Part (i): The total number of length-7 strings with no repeated characters is: $93,963,542,400$

2. Part (ii): The total number of length-6 strings with no repeated characters and the first character not being a special character is: $2,487,270,240$

Would you like further clarification or a breakdown of any part of the solution?

Related Questions:

1. How does excluding the first character from special characters change the calculation?
2. What would be the number of passwords if repetition of characters was allowed?
3. How does the inclusion of a case-sensitive letter set (e.g., lowercase and uppercase) affect the total counts?
4. What is the factorial formula used to derive the results in these cases?
5. How can we adapt the calculations for a different length of the password?

Tip:

To reduce errors in combinatorial problems, clearly identify constraints and always validate that your formula matches the problem's requirements.