What is the exponential generating function of the sequence $a_n$ = number of passwords with at least one capital letter, one number and one special character.
I know that if I were to restrict specific letters, such as "must have at least 1 a", I can merely change the function for a with the remaining having a function of $e^x$ but I'm not sure how to do at least one letter or one digit etc with exponential generating functions
There are $26$ lower case letters, $26$ upper case, $10$ digits, $34$ special characters, making $96$ in all. So $$a_n=96^n-70^n-86^n-62^n+60^n+36^n+52^n-26^n$$ by application of the inclusion-exclusion principle. Then the exponential generating function is $$\sum_0^{\infty}{a_nx^n\over n!}=e^{96x}-e^{70x}-e^{86x}-e^{62x}+e^{60x}+e^{36x}+e^{52x}-e^{26x}$$ This factors as $$e^{96x}(1-e^{-26x})(1-e^{-10x})(1-e^{-34x})$$