document.write( "Question 948972: Is there a difference in number of combinations for the two passwords:\r
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Algebra.Com's Answer #579435 by Edwin McCravy(20055)\"\" \"About 
You can put this solution on YOUR website!
Is there a difference in number of combinations for the two passwords:\r
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document.write( "[You should not use the word \"combination\" to refer to these passwords,\r\n" );
document.write( "because \"combination\" is not used the same way in mathematics as it is \r\n" );
document.write( "used in everyday discourse.]\r\n" );
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document.write( "I will assume that repetitions are allowed, since you did not say anything\r\n" );
document.write( "to rule them out.\r\n" );
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document.write( "We use the indirect method.\r\n" );
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document.write( "First we calculate the number of 8-character passwords that can be made from\r\n" );
document.write( "any of the 26 small letters (a,...,z} plus the 26 capital letters {A,....Z},\r\n" );
document.write( "plus the 10 digits {0,...9}, plus the 7 characters {!,\",§,$,%,&,/}.\r\n" );
document.write( "That's 26+26+10+7 = 69 characters.\r\n" );
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document.write( "That's 69^8 = 513798374428641 passwords  \r\n" );
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document.write( "Then we subtract the passwords that we do not want to use:\r\n" );
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document.write( "1. Passwords that do not use any of the 26 small letters, which is\r\n" );
document.write( "(69-26)^8 = 43^8 = 11688200277601 passwords.\r\n" );
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document.write( "2. Passwords that do not use any of the 26 capital letters, which is\r\n" );
document.write( "also (69-26)^8 = 43^8 = 11688200277601 passwords.\r\n" );
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document.write( "3. Passwords that do not use any of the 10 digits, which is\r\n" );
document.write( "(69-10)^8 = 59^8 = 146830437604321 passwords.\r\n" );
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document.write( "4. Passwords that do not use any of the 7 characters {!,\",§,$,%,&,/}, \r\n" );
document.write( "which is (69-7)^8 = 62^8 = 218340105584896 passwords.\r\n" );
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document.write( "Subtracting the unwanted passwords from the 69^8 \r\n" );
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document.write( "69^8 - 43^8 - 43^8 - 59^8 - 62^8 = 125251430684222 passwords A\r\n" );
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document.write( "This time we use the direct method.\r\n" );
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document.write( "1. Passwords that use only small letters, capital letters and digits:\r\n" );
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document.write( "That's (26+26+10)^8 = 62^8 = 218340105584896 passwords\r\n" );
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document.write( "2. Passwords that use only small letters, capital letters and those \r\n" );
document.write( "special 7 characters:\r\n" );
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document.write( "That's (26+26+7)^8 = 59^8 = 146830437604321\r\n" );
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document.write( "3. Passwords that use only small letters, digits and those special 7\r\n" );
document.write( "characters:\r\n" );
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document.write( "That's (26+10+7)^8 = 43^8 = 11688200277601 passwords.\r\n" );
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document.write( "4. Passwords that use only capital letters, digits and those special 7\r\n" );
document.write( "characters:\r\n" );
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document.write( "That's also (26+10+7)^8 = 43^8 = 11688200277601 passwords.\r\n" );
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document.write( "62^8 + 59^8 + 43^8 + 43^8 = 388546943744419 passwords B\r\n" );
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document.write( "Final answer: Yes, there is a difference because\r\n" );
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document.write( "69^8 - 43^8 - 43^8 - 59^8 - 62^8 = 125251430684222 passwords A\r\n" );
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document.write( "is not the same as\r\n" );
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document.write( "62^8 + 59^8 + 43^8 + 43^8 = 388546943744419 passwords B\r\n" );
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document.write( "Edwin
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