Need help with this pigeonhole principle based problem:
Assume computer passwords are between $6$ and $8$ characters long. Each character can be either an upper case letter, a lower case letter or a digit. Assume each password must have at least one upper case letter, at least one lower case letter and at most one digit.
Let $S$ be the set of passwords. What is the minimum cardinality of $S$ to guarantee that at least $2$ passwords in $S$ have the same number of lower case letters?