Final answer:
The number of valid 11-character passwords that do not repeat any character and include X, Y, and Z is calculated by using permutations. The correct formula is 36!/(36-11)!, which accounts for the decreasing number of choices as each character is selected.
Step-by-step explanation:
The question is asking for the number of valid passwords with a length of 11 characters, only using capital letters (A-Z) and digits (0-9), where each character is not repeated and contains the letters X, Y, and Z. To solve this, we need to calculate the number of ways to arrange the remaining 8 characters after X, Y, and Z are already chosen to be in the password.
Since there are 26 capital letters and 10 digits, we have 36 possible characters. However, three letters have been used, leaving us with 33 characters to choose from for the remaining 8 spots. Moreover, no character may be repeated. We can calculate the number of these permutations by the formula 33!/(33-8)! or 33!/25! This represents selecting 8 distinct characters from the remaining pool of 33 without repetition.
Therefore, the correct answer is c. 36!/(36-11)! since initially, we can choose from all 36 characters, and for each subsequent character, we have one less choice, not reusing any of the chosen characters including X, Y, and Z. Hence, the 11-character password count is the number of permutations of 36 characters taken 11 at a time without repetition.