Suppose a four-character password must be formed using only the digits 0-9, the

Question

Suppose a four-character password must be formed using only the digits 0-9, the capital letters A-F, and the lowercase Greek letters alpha, beta, gamma, and omega. It is acceptable for the password to repeat characters, and the order of characters in a password matters. How many such passwords ... are there in total? don't repeat any character at all? start with a capital letter, have digits in the middle two positions, and end with a Greek letter? start with A and end with omega. have alpha in exactly two positions? have Greek letters in exactly two positions? have Greek letters in exactly two positions, which happen to be adjacent positions? have Greek letters in at least two positions? have no Greek letters? consist only of digits? have at least one digit?

Explanation / Answer

Dear Student Thank yu for using Chegg !! Given Some characters as follows 0,1,2,3,4,5,6,7,8,9,A,B,C,D,E,F,,,, a) Total Number of Passwords For each position we have 20 options => Solution is 20^4 160000 b) No characters repeated => 20X19X18X17 => 116280 c) Start with Capital, Digits in middle two and end with greek => 6X10X10X4 => 2400 d) Start with A and end with => 20X20 400 e) Have in exactly two positions Now selecting position for no of selctions are 4C2 i.e. 6 In the rest positions (2 nos) we can have any of the rest 19 characters => 2166 f) Have greek letters in exactly two positions Now there are two possible cases i.e. either we have same greek letter in two positions or we have different greek letter in two positions i. Same greek letter combination with section of the two places out of 4 available 4C1 X 4C2 24 ii) Different greek letter combination with section of the two places out of 4 available 4C2 X 4C2 X2! 72 iii) Permutation for the rest two positions we have 16 options for the other two positions 256 Therefre overall solution is ( i) + ii) )*iii) 24576 g) Have greek letters in exacty two positions which happen to be adjacent position The problem is same as previous problem only difference being in selection of two places where no of combination are reduced from 6 to 3 i) 12 ii) 36 iii) 256 Solution is 12288 h) Have greek letters in atleast two positions This problem is solved by subtracting the case when all 4 characters of password are either not greek or one of the charater is greek from total number of password possibilities i) All four characters are not greek 65536 ii) Only One charatcer is greek 4C1 X 4C1 X 16X16X16 65536 => 28928 i) Have no greek letters => 65536 j) Consist only of digits 10000 k) Have atleast one digit Subtracting no of cases where we have no digit in password from total permutation shall give the number of cases with atleast one digit 150000

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