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Math Help - Eight character passwords

  1. #1
    Member oldguynewstudent's Avatar
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    Eight character passwords

    How many eight character passwords are there if each character is either an uppercase letter A-Z, a lowercase letter a-z, or a digit 0-9, and where at least one character of each of the three types is used?

    First how many total 8 character passwords are possible? 62^8

    Second how many 8 character passwords without any digits are possible? 52^8

    Third how many 8 character passwords without any lowercase letters are possible? 36^8

    Finally how many 8 character passwords without any uppercase letters are possible? 36^8

    The answer should be 62^8 - 52^8 - 36^8 - 36^8; correct?
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  2. #2
    MHF Contributor undefined's Avatar
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    Quote Originally Posted by oldguynewstudent View Post
    How many eight character passwords are there if each character is either an uppercase letter A-Z, a lowercase letter a-z, or a digit 0-9, and where at least one character of each of the three types is used?

    First how many total 8 character passwords are possible? 62^8

    Second how many 8 character passwords without any digits are possible? 52^8

    Third how many 8 character passwords without any lowercase letters are possible? 36^8

    Finally how many 8 character passwords without any uppercase letters are possible? 36^8

    The answer should be 62^8 - 52^8 - 36^8 - 36^8; correct?
    I believe you should apply the principle of inclusion-exclusion in this case; because for example there are some passwords that contain neither numbers nor uppercase letters, and you subtracted these out twice.

    So the additional things to count would be

    Fifth how many 8 char passwords without any any digits or lowercase? 26^8

    Sixth how many 8 char pass without any digits or uppercase? 26^8

    Seventh how many 8 char pass without any lowercase or uppercase? 10^8

    Final answer:

    62^8 -52^8 - 36^8 - 36^8 + 26^8 + 26^8 + 10^8
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  3. #3
    Member oldguynewstudent's Avatar
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    Quote Originally Posted by undefined View Post
    I believe you should apply the principle of inclusion-exclusion in this case; because for example there are some passwords that contain neither numbers nor uppercase letters, and you subtracted these out twice.

    So the additional things to count would be

    Fifth how many 8 char passwords without any any digits or lowercase? 26^8

    Sixth how many 8 char pass without any digits or uppercase? 26^8

    Seventh how many 8 char pass without any lowercase or uppercase? 10^8

    Final answer:

    62^8 -52^8 - 36^8 - 36^8 + 26^8 + 26^8 + 10^8
    I see your point. I haven't gotten to the inclusion-exclusion part of the book yet. I'm trying to get a head start on my summer courses since I'm taking a graduate level combinatorics and undergrad applied statistics at the same time and I'm a little apprehensive.
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