Is there a difference in number of combinations for the two passwords:
PASSWORD A
8 characters, at least one character out of each of the 4 groups
a-z, A-Z, 0-9 and !"§$%&/
[You should not use the word "combination" to refer to these passwords,
because "combination" is not used the same way in mathematics as it is
used in everyday discourse.]
I will assume that repetitions are allowed, since you did not say anything
to rule them out.
We use the indirect method.
First we calculate the number of 8-character passwords that can be made from
any of the 26 small letters (a,...,z} plus the 26 capital letters {A,....Z},
plus the 10 digits {0,...9}, plus the 7 characters {!,",§,$,%,&,/}.
That's 26+26+10+7 = 69 characters.
That's 69^8 = 513798374428641 passwords
Then we subtract the passwords that we do not want to use:
1. Passwords that do not use any of the 26 small letters, which is
(69-26)^8 = 43^8 = 11688200277601 passwords.
2. Passwords that do not use any of the 26 capital letters, which is
also (69-26)^8 = 43^8 = 11688200277601 passwords.
3. Passwords that do not use any of the 10 digits, which is
(69-10)^8 = 59^8 = 146830437604321 passwords.
4. Passwords that do not use any of the 7 characters {!,",§,$,%,&,/},
which is (69-7)^8 = 62^8 = 218340105584896 passwords.
Subtracting the unwanted passwords from the 69^8
69^8 - 43^8 - 43^8 - 59^8 - 62^8 = 125251430684222 passwords A
PASSWORD B
8 characters, at least one character out of three of the same 4 groups.
This time we use the direct method.
1. Passwords that use only small letters, capital letters and digits:
That's (26+26+10)^8 = 62^8 = 218340105584896 passwords
2. Passwords that use only small letters, capital letters and those
special 7 characters:
That's (26+26+7)^8 = 59^8 = 146830437604321
3. Passwords that use only small letters, digits and those special 7
characters:
That's (26+10+7)^8 = 43^8 = 11688200277601 passwords.
4. Passwords that use only capital letters, digits and those special 7
characters:
That's also (26+10+7)^8 = 43^8 = 11688200277601 passwords.
62^8 + 59^8 + 43^8 + 43^8 = 388546943744419 passwords B
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Final answer: Yes, there is a difference because
69^8 - 43^8 - 43^8 - 59^8 - 62^8 = 125251430684222 passwords A
is not the same as
62^8 + 59^8 + 43^8 + 43^8 = 388546943744419 passwords B
Edwin