One of the most common tasks in the online world is to write an account on a website of interest.Typically, a part of this task is to show a password.While this can sometimes be a nuisance, there is often a real security rationale behind the requirement.

1) An easy way to increase password security is to require that each password have at least a certain minimum length.This certainly helps increase the number of possibilities.

a) Assuming that the password is NOT case-sensitive and that letters can be used more than once, how many 5 letter passwords are possible?(6 points)

b) Investigate how this changes as the minimum length is increased by finding the number of possible 6, 7 and 8 letter passwords under the same assumptions.(6 points)

2) Allowing (or requiring) users to use numerical digits (including 0) and one of 33 "special characters" dramatically increases the number of possible passwords.Furthermore, passwords often are case-sensitive, effectively doubling the size of the alphabet by defining 52 distinct letter characters.

Assuming that passwords are case-sensitive and can include numerical digits and special characters, how many possible 5, 6, 7 and 8 character passwords can be created?(8 points)

3) Allowing digits and special characters creates an enormous number of passwords.However, computers can check passwords very quickly.Let's say that a computer can check 100 million passwords per second.

a) How long (in minutes) would it take a computer to check every possible 5 character password? (8 points)

b) How long (in days) would it take a computer to check every possible 8 character password? (8 points)

4) Question 2 asked you to compute all possible 7 character passwords.However, it seems more likely that most people will choose actual words for their password.According to the websitewordfinder.yourdictionary.com, the English language has 8887 five-letter words and 15,727 six-letter words.

Assume that a password must be 7 characters and include at least 1 digit and at least 1 special character.If people form their password by taking an English word of exactly 5 or 6 letters (assume that this password is not case-sensitive) and then adding (in either order) a digit and a special character, how many passwords would be possible? (14 points)

(Hint: First find the number of passwords with 5 letters followed by a digit and a special character, then exchange the order of the last two characters.Then do the same for 6 letter words).