How many different passwords are there that contain only digits and lower-caseletters and satisfy the given restrictions?(iii) Length is 7 and the password must contain at least one digit.

36⁷- 26⁷ = 70332353920 passwords.

Step-by-step explanation:

Number of digits 0-9 = 10

Number of lowercase letters a-z = 26

We have to find different passwords of length 7 that contain only digits and lower-case letters.

Total number of characters = 26 lowercase letters + 10 digits

                                              = 36

The total number of length 7 passwords without restrictions = 36⁷

                                                                                                    = 78364164096

The number of length 7 passwords with no digits (only lowercase letters)

   =  26⁷

   = 8031810176

Now the restriction here is that the password must contain at least 1 digit. This can be computed by subtracting the number of length 7 passwords with no digits from total number of length 7 passwords.

Number of length 7 passwords with restriction (at least one digit) = 36⁷- 26⁷

       = 36⁷- 26⁷

       = 78364164096 - 8031810176

       = 70332353920

So the number of different passwords that contain only digits and lower-case letters and satisfy the given restrictions that length is 7 and the password must contain at least one digit = 70332353920