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Math Help - counting + password question

  1. #1
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    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
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  2. #2
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    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.
    A password must start with a letter and have at least one digit.
    How many passwords are possible?
    We will count the number of 6-, 7- and 8-character passwords and add.


    6-character passwords

    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]


    7-character passwords

    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]


    8-character passwords

    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}

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