A computer system uses passwords that are exactly seven characters and each char

Question

A computer system uses passwords that are exactly seven characters and each character is one of the 26 letters (a-z) or 10 integers (0-9). Uppercase letters are not used. Answer the following questions under two different scenarios, [1]-the characters can be duplicated (used more than once in the same password), [2]-the characters can be used once. How many passwords are possible? If a password consists of exactly 6 letters and one number: In how many ways the number can be positioned in the password? How many passwords are possible? If a password consists of 5 letters and two numbers, how many passwords are possible? In how many ways the two numbers can be positioned in the password? How many passwords are possible? Generate a function that gives the number of possible passwords formed by n characters with i letters and j numbers (i+j=n, and n=1,2,3....) What is the probability to guess a password when you are given all its 7 characters (not ordered)? What is the probability to guess a password that has only two missing characters? Answer the above questions (1-6) under the assumption that a character is used once in the password, i.e, all the characters forming the password are different.

Explanation / Answer

there are all total 7 characters which can either be filled by 26 letters or 10 integers.

1. so each of the 7 characters can be any one of the 26+10=36 characters

so under scenario [1] [the characters can be duplicated]

so the first character can be any one of the 36 characters, so as the second one, so as the third one and so on.

hence possible number of passwords=36*36*36*36*36*36*36=367 [answer]

under scenario [2] [the characters can be used only once]

the first character can be any one of the 36 characters, the second one can be any one of the remaining 35 characters, the third one can be any one of the remaining 34 characters and so on

hence possible number of passwords=36*35*34*33*32*31=1402410240 [answer]

2. now it is said that the password can have only 6 letters and 1 integer

i) so the number can be positioned in any position out of 7 positions in 7C1=7 ways [answer]

ii) under scenario [1] [the characters can be duplicated]

out of the 7 characters the 6 characters which are letters , each of them can be any one of the 26 letters

and the remaining one which is the number can be any one of the 10 numbers. and the number can be positioned in any position out of 7 positions in 7C1=7 ways

so total possible passwords=7*266*10 [answer]

onder scenario [2] [the characters can be used only once]

out of the 7 characters the 6 characters which are letters , the first one can be any one of the 26 letters,second one can be any one of the remaining 25 letters and so on

and the remaining one which is the number can be any one of the 10 numbers. and the number can be positioned in any position out of 7 positions in 7C1=7 ways

so total possible passwords=7*26*25*24*23*22*21*10 [answer]

3. now it is said that the password can have only 5 letters and 2 integers

i) so the 2 numbers can be positioned in any position out of 7 positions in 7C2=21 ways [answer]

ii) under scenario [1] [the characters can be duplicated]

out of the 7 characters the 5 characters which are letters , each of them can be any one of the 26 letters

and the remaining two which are the numbers can be any one of the 10 numbers. and the numbers can be positioned in any position out of 7 positions in 7C2=21 ways

so total possible passwords=21*265*102 [answer]

onder scenario [2] [the characters can be used only once]

out of the 7 characters the 5 characters which are letters , the first one can be any one of the 26 letters,second one can be any one of the remaining 25 letters and so on

and the remaining two which are the numbers the first one can be any one of the 10 numbers.the second one can be any one of the remaining 9 numbers and the number can be positioned in any position out of 7 positions in 7C2=21 ways

so total possible passwords=7*26*25*24*23*22*10*9 [answer]

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