Is the sum of permutations of disjointed sets larger than the count of permutations in their union? My wife and I are having a bit of a disagreement.
Concerning eight-digit passwords like the kind most secure websites require, she believes you could generate more unique password combinations by using only one of the following sets: one lowercase or one uppercase or one symbol or one numeral. In other words, if you created a password using only lowercase letters with no other conventions. e.g.: jtpdlrkc or JTPDLRKC or 29456014 or #^@%#&(%.
I have asserted that, by using at least one each of the aforementioned sets, a person can generate FAR, FAR more unique password combinations, thereby making the password much more difficult for someone to hack. e.g. gy7HU*j9 or $F6ms38@.
She thinks I'm making this up. I need numbers to prove my point.
Anyone interested in running with this one?

There is a context in which the wife is right, which may explain her intuition.  Suppose we split the permitted symbols into two sets, $S$ and $T$, of size $s=|S|$ and $t=|T|$.  Suppose further that our password must be exactly two letters long.
The wife's strategy yields $s^2+t^2$ passwords, either two from $S$ or two from $T$.  The OP's strategy yields $2st$ passwords, either one from $S$ then one from $T$, or the other way.  
The wife's strategy will give at least as many passwords as the OP's, because $(s^2+t^2)-2st=(s-t)^2\ge 0$.  Indeed,  if $s\neq t$, then the wife's strategy gives strictly more passwords.
