Question 176545
OK, so a password can be 6, 7, or 8 characters long.
The first character has to be a letter (could be upper or lowercase), so that makes 52 (2x26) possible choices. 
One of the remaining slots must be a digit with 10 possible choices (0-9). 
All of the other remaining slots (total of 4,5,6) can either be a digit, an upper case letter, or a lower case letter with 10+2x26=62 possible choices.
Each slot choice is independent so the total number of passwords is the product of each slot’s number of choices.

So for a 6 character password, there is the first letter slot, the digit slot and then there are 4 remaining slots.
N[6]=52*10*62*62*62*62=520*62^4 possible choices.
For a 7 character password, there are 5 remaining slots.
N[7]=52*10*62*62*62*62*62=520*62^5 possible choices.
And finally for an 8 character password, there are 6 remaining slots,
N[8]=52*10*62*62*62*62*62*62=520*62^6 possible choices.
The total number is then the sum of 
N[6,7,8]=520*(62^4+62^5+62^6)=3.002E^12
That's a little over 3 trillion possible passwords.
Just to put some sense to that number, say you wanted to break into someone's account.
If you entered 1 password every second, after 1 year of doing it (with no breaks) you'd have only entered 3 million passwords or .001% of the total possible passwords.