# How does increasing minimum password length affect the size of password space?

I understand that when there are a possible characters and passwords must be of length n there are an possible passwords.

But usually passwords can vary in length between some minimum and maximum length. If the minimum length is k and the maximum length is l, would the number of possible passwords actually be ak + ak+1 + ak+2 + ... + al?

And, if so, how does increasing the minimum length requirement (say, from k to k + s) reduce the total number of possible passwords?

Yes, you've correctly computed the number of passwords provided `a` is the number of possible characters, they are all single-byte (or you compute length based on Unicode codepoints), and there are no composition requirements. For many sites, `a` is 94, the number of non-whitespace printable ASCII characters, or 95, which includes space. Unfortunately, too many sites choose worse options, and very few sites allow Unicode characters because they require normalization to be functionally useful.
It is also true that increasing the minimum password length actually reduces the number of possible passwords. To compute the difference, you compute your sequence with `k` and then compute it with `k + s`, and then you subtract the latter from the former to get the reduction in the space.
However, all of that being said, there are a lot of practical constraints here that affect the attack surface. First, `l` should practically be a reasonably large number, such as 128 or 1024, so that people can use passphrases or long generated passwords. Even if you are using bcrypt, `l` can be as large as 72. With that constraint, the number of possible passwords is very large (the entropy in a truly random 72-character password out of a 94-character set is more than 471 bits), and thus the number of passwords is not practically reduced by very much. Note that using the theoretical minimum energy to store 2^128 bits is enough to boil the world's oceans, so physics tells us it is very unlikely that someone will be able to examine a meaningful number of those passwords.