Combinatorics password problem

Question

a 7-character password can have upper case letters, lower case letters, and numerical digits. How many passwords are possible if it must start with a number and end with a lowercase letter (repetitions allowed)?

Answer 1

There are 26 upper case letters, 26 lower case letters, and 10 numbers,

for a total of 62 possible characters.

The first position can be chosen in 10 ways.

The last position can be chosen in 26 ways.

Each of the remaining 5 positions can be chosen in 62 ways.

This gives a total of 10×62⁵×26 possible passwords.

The constraint on the first position guarantees a digit,

the constraint on the last position guarantees a lower case letter,

but the above count includes passwords without any upper case letter.

Therefore from the above count we need to subtract those passwords can contain

only lower case letters and digits.

By the same reasoning as above there are 10×36⁵×26 such combinations.

Therefore the number of allowed passwords is

10×62⁵×26 - 10×36⁵×26 = 222,473,330,560