TITLE: How many ways can you create a password of 10 characters long that has at least one lowercase letter (a-z) and at least one number ($0-9$)?
QUESTION [5 upvotes]: Suppose you want to generate a password using ASCII  characters ($128$ characters.)How many ways can you create a password of 10 characters long that has at least one lowercase letter (a-z) and at least one number ($0-9$)?
** MY ANSWER: $26*10*(128)^{10}$**
I don't understand why this is wrong? Can anybody help? Thanks!

REPLY [3 votes]: The easiest way is to count all the passwords, then deduct those that don't meet the requirement.  There are $128^{10}$ total passwords, ignoring the requirement for a lower case letter and a number.  Of those $(128-26)^{10}$ have no lower case letter and $(128-10)^{10}$ have no number.  But we have deducted twice those that have no lower case letter and no digit-an example of the inclusion-exclusion principle.  So we add them back in-there are $(128-36)^{10}$ of those.  So the final answer is $128^{10}-102^{10}-118^{10}+92^{10}=578747468760442009600$ per Alpha$