Password length is 5 or 6 characters. Characters allowed are $a-z$, $A-Z$, $0-9$.
The password must contain at least one lowercase letter, at least one uppercase letter and at least one digit.
Use the inclusion, exclusion principal. $|\{lud\}| - |\{lu\}|-|\{ld\}|-|\{ud\}| + |\{l\}|+|\{u\}|+|\{d\}|$
$$C= (26\!+\!26\!+\!10)^5(26\!+\!26\!+\!10\!+\!1)\!-\!(26\!+\!26)^5(26\!+\!26\!+\!1)\!- 2(26\!+\!10)^5(26\!+\!10\!+\!1)\!+\!2(26)^5(26\!+\!1)\!+\!(10)^5(10\!+\!1)$$
$$C= (62)^5(63) - (52)^5(53) - 2(36)^5(37) + 2(26)^5(27)+(10)^5(11)$$
$$C= 57,\!716,\!368,\!416 - 20,\!150,\!813,\!696 - 4,\!474,\!497,\!024 + 641,\!594,\!304+1,\!100,\!000$$
$$C= 57,\!716,\!368,\!416 -24,\!625,\!310,\!720 + 642,\!694,\!304$$
$$C= 33,\!733,\!752,\!000$$
but it doesn't match this:
$(26∗26∗10∗62∗62)+(26∗26∗10∗62∗62∗62)=1,637,082,720$
That only counts passwords starting with one upper case, one lower case, and one digit, --in that order-- followed by 2 or 3 more symbols from any group.
It does not count, for instance, "$\mathrm{HeLL0}$"
We might as well assume case doesn't matter - if it does, you just need to adjust the numbers slightly. The total number of strings of characters is, as you pointed out, $36^4$. However, not all of these are passwords. If all characters are letters ($26^4$ possibilities) or all characters are numbers ($10^4$ possibilities, none of which doubles up with a case where all characters are letters) then our string is not a password. In all other cases the string IS a password. Hence there are
$$36^4-(26^4+10^4)=1\ 212\ 640$$
possible passwords. If the password is case sensitive, the answer becomes
$$62^4-(52^4+10^4)=7\ 454\ 720.$$
Best Answer
All you would have to do would be to take the number of possible passwords with uppercase and lowercase combined, then subtract the number of all uppercase and all lowercase combinations.
You would end up with $52^5 - 2*26^5$