Question

. A computer service recently added a new six-character password to their login system. The first character
of the password has to be a letter and must be uppercase. The last character had to be a digit (0 to 9). The
other four characters could be either digits or letters, and those letters could be either uppercase or
lowercase, in other words the password is “case sensitive.” How many different passwords could be created
if letters and digits may be repeated?

Answer 1

First come up with the answers for:
a) How many upper case letters are there in the English alphabet?
b) How many digits are there?
c) What is the total of: uppercase letters + lowercase letters + digits?

Answer 2

Let's see if i get it right.
First character is A to Z, so total 26 possible values.
Then we have 4 characters that could be a to z, A to Z or 0 - 9. that is 26 for uppercase, 26 for lowercase and 10 for digits so total 26 + 26 + 10 = 62 possible values
Then we have last character that could be only a digit, so 10 possible values.
Basically that means that for each one of the 26 possible first letters, there are 62 possible values for second characters.
So for example, for only 2 characters of the pass there are
\[
\text{First_letter_possiblities} \cdot \text{Second_letter_possibilites} = 26 \cdot 62 = 1612
\]
Now, for each one of those 1612 combination... there are another 62 possible combinations for the third character.
In short:
\[
{\small \text{Possible passwords} = \text{1st_allowed_chars} \cdot \text{middle_allowed_chars} \cdot \text{last_allowed_chars} } \\
\text{1st_allowed_chars} = 26 \\
\text{middle_allowed_chars} = 62 \cdot 62 \cdot 62 \cdot 62 = 62^4 \\
\text{last_allowed_chars} = 10\\
\text{Possible passwords} = 26 \cdot 62^4 \cdot 10 = 260 \cdot 14,776,336 = 3,841,847,360
\]
If i got it right