Question: A sequence of exactly 8 symbols, which contains exactly one lower-case letter from ${a, . . . , z}$, exactly one upper-case letter from ${A, . . . , Z}$ and at least one from ${0, 1, . . . , 9}$.
I used rule of product and assumed:
Choosing one letter from $26$ small letters for 8 positions: $(26*8)$
Choosing one letter from $26$ capital letters for 7 positions: $(26*7)$
Choosing at least $1$ digit from $0$ to $9$: $(10^6)$
I worked it out to be $26^8 *26^7-10^6$, not sure if this right.
(Assuming $26$ letter alphabet and $10$ digits from $0$ to $9$. It doesn't specifically mention this but it also asks for the sequences to be unique. Note: Sorry if this isn't in the right format, been a while since I was on here.)
If I understand you correctly, you require a password with exactly one lower case letter, exactly one upper case letter and digits on all positions.
First, lets choose the place of the lower case letter. There are exactly $8$ positions where it can be and there are $26$ ways to chose this letter. Now, there are $7$ positions left and one of the $26$ uppercase letters is in one of them. And all the other symbols are digits (there is $10$ ways to chose each of them). Thus we get $8*7*26^2*10^6 = 2^{11}*5^6*7*13^2$ different ways to construct such a password.