1. ## counting + password question

Heres the question im stuck on...
A password with between 6 and 8 characters, each can be lowercase, uppercase, or any digit 0-9. A password must start with a letter and have atleast one digit. ex. How many passwords are possible?

Heres what I m thinking, but I have a feeling its more difficult than this...

for the first letter you would have 52 choices (upper plus lower) and also you require a digit for one (10 choices) the rest it could be any 62 choices (digits plus letters)

52 x 10 x 62 x 62 x 62 x 62 x 62 x 62... doesnt seem correct.. help if you can

2. Hello, zackgilbey!

You are correct . . . it's much more difficult.

A password with between 6 and 8 characters, each can be
lowercase, uppercase, or any digit 0-9.
We will count the number of 6-, 7- and 8-character passwords and add.

The first is a letter, 52 choices.. The other five have 62 choices.
. . With no other restrictions, there are: . $52\cdot62^5$ possible passwords.

But it must contain at least one digit.

How many passwords have no digits (all letters)?
There are: . $52^6$ with no digits.

Hence, there are: . $52\cdot62^5 - 52^6$ six-character passwords with at least one digit. [1]

The first is a letter, 52 choices.. The other six have 62 choices.
. . With no other restrictions, there are: . $52\cdot62^6$ possible passwords.

But it must contain at least one digit.

How many passwords have no digits (all letters)?
There are: . $52^7$ with no digits.

Hence, there are: . $52\cdot62^6 - 52^7$ seven-character passwords with at least one digit. [2]

The first is a letter, 52 choices.. The other seven have 62 choices.
. . With no other restrictions, there are: . $52\cdot62^7$ possible passwords.

But it must contain at least one digit.

How many passwords have no digits (all letters)?
There are: . $52^8$ with no digits.

Hence, there are: . $52\cdot62^7 - 52^8$ eight-character passwords with at least one digit. [3]

These numbers are beyond the capacity of my calculator,

. . but the sum of [1], [2] and [3] seems to be about: . $1.316176398 \times 10^{14}$