We are given a password-based authentication system that requires the following constraints:

• The password must be 15 characters long.
• It must contain exactly:
  • 2 digits (chosen from the 10 digits: 0, 1, 2, ..., 9),
  • 2 special characters (chosen from the 6 special characters: {!, @, #, $, %, &}),
  • The remaining 11 characters must be upper or lower case letters (chosen from 26 upper case and 26 lower case letters, so 52 letters total).

Step-by-step calculation:

1. Select 2 positions for the digits:

   • Out of the 15 positions in the password, we need to choose 2 positions for the digits.
   • The number of ways to select 2 positions out of 15 is given by the combination formula $\binom{n}{k}$: $\binom{15}{2} = \frac{15 \times 14}{2 \times 1} = 105$

2. Select 2 positions for the special characters:

   • After assigning 2 positions for the digits, there are 13 remaining positions.
   • We need to select 2 positions for the special characters.
   • The number of ways to choose 2 positions out of 13 is: $\binom{13}{2} = \frac{13 \times 12}{2 \times 1} = 78$

3. Assign digits to the chosen positions:

   • Each digit can be any of the 10 digits (0 to 9).
   • The number of ways to assign digits to the 2 positions is: $10 \times 10 = 100$

4. Assign special characters to the chosen positions:

   • Each special character can be any of the 6 special characters.
   • The number of ways to assign special characters to the 2 positions is: $6 \times 6 = 36$

5. Assign letters to the remaining 11 positions:

   • The remaining 11 positions must be filled with letters (upper or lower case).
   • The number of ways to assign letters to these 11 positions is: $52^{11}$

6. Total number of passwords:

   • Now, we multiply all these possibilities together to get the total number of valid passwords: $\text{Total passwords} = \binom{15}{2} \times \binom{13}{2} \times 10^2 \times 6^2 \times 52^{11}$ Substituting the values we computed: $\text{Total passwords} = 105 \times 78 \times 100 \times 36 \times 52^{11}$

Thus, the total number of such passwords is:

$105 \times 78 \times 100 \times 36 \times 52^{11}$

This simplifies to approximately $1.693 \times 10^{22}$ distinct passwords.

Would you like more details or have any questions?

Here are some related questions you might explore:

1. What if the password could contain 1 to 3 digits, rather than exactly 2?
2. How would the total number of passwords change if only lowercase letters were allowed?
3. What is the probability of randomly generating a valid password that meets the above conditions?
4. How does the length of the password affect the security of the system?
5. How many passwords would be possible if the number of special characters increased from 6 to 10?

Tip: When calculating combinations, remember that choosing positions and assigning values are separate steps—calculate each step carefully!