A computer system uses passwords that contain exactly seven characters, and each

ID: 3298504 • Letter: A

Question

A computer system uses passwords that contain exactly seven characters, and each character is

1 of the 26 lowercase letters (a–z) or 26 uppercase letters (A–Z) or 10 integers (0–9). Let

denote the set of all possible passwords, and let A and B denote the events that consist of

passwords with only letters or only integers, respectively.

Determine the number of passwords in each of the following events.

(a) (b) A (c) A B

(d) Passwords that contain at least 1 lowercase letter

Determine the probability for each of the following:

(e) Password contains at least 1 lowercase letter given that it contains only letters

(f) Password contains only odd numbers given that it contains all numbers

(According to Chegg policy, only four subquestions will be answered. Please post the remaining in another question)

(a) : There are a total of 26 lowercase letters, 26 uppercase letters and 10 integers - 62 in all.

Since each password has 7 characters, total number of passwords = 627

(b) A : There are a total of 26 lowercase letters and 26 uppercase letters - 52 in all.

Since each password has 7 characters, total number of passwords = 527

A B : According to De-Morgan's law this is (A U B)' and the number of passwords in A U B = 527 + 107 as the two events are mutually exclusive.

=> Number of passwords in (A U B)' = 627 - 527 - 107

(d) Passwords that contain at least 1 lowercase letter

Firstly let us find the number of passwords that don't contain any lowercase letters at all.

There are 26 uppercase letters and 10 digits - 36 in all.

Total number of passwords without any lowercase letters = 367

=> Total number of password with atleast one lowercase letter = 627 - 367.

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