Final answer:
The number of different passwords with the first digit as 6 and no repeated digits depends on the length. For a four-digit password, we multiply the options for each digit, which would be 1 x 9 x 8 x 7. For a ten-digit password, we would use the same pattern up to the tenth digit.
Step-by-step explanation:
The question pertains to the number of unique passwords that can be created with the condition that the first entry is the digit 6 and no other digit is repeated in the password. Assuming we are creating a password of a certain length, let's say a four-digit password, we have the following scenario: the first digit is fixed as 6, leaving us with 9 options (digits 0-5 and 7-9) for the second digit, 8 options for the third, and 7 for the fourth. To find the total number of possible passwords, we would multiply these possibilities together: 1 x 9 x 8 x 7. For a longer password, we would continue this pattern until we have allocated all available digits.
If we do not specify the password length, then for a system that uses digits only, the maximum length of a password that starts with 6 and does not repeat any digits would be 10, since there are 10 different digits (0-9). The principle of counting remains the same: the first digit is fixed as 6, and the remaining digits are selected from the remaining 9 unique digits, then 8, and so on, down to 1.