Question

Passwords on an ancient computer are required to be 4-5 characters long and made up of lower case letters only (a..z : there are 26 possibilities). Please answer each of these questions below, giving numerical answers. You should also explain how you got each answer, for full or partial credit.

a) Assume that letters may NOT be repeated, and passwords are 4-5 letters long. How many passwords are possible?

b) Assume that letters MAY be repeated, and passwords are 4-5 letters long. How many passwords are possible?

c) Assuming that letters MAY be repeated, what proportion of the total number of passwords in fact *do* have at least one letter used more than once?

d) Letters may be repeated but the following extra rules exist:

• The password cannot be 4 or 5 identical letters like xxxx or bbbbb
• The password cannot be on a list of 71 common 4-5 letter words (none of them are like xxxx or bbbbb!)

How many passwords are possible now?

Note: since this is a practice test question, the answers can be seen in the question comment after you turn in your quiz. Canvas can't grade "essay" type questions for correctness, so you will get a 0 grade whatever you write.

Answer 1

a)

Case1: Password is four letters long

The first letter has 26 choices. The second letter has 25 choices, since the letters cannot be repeated. Similarly, the third letter has 24 choices, and the fourth letter has 23 choices.

Total number of password possible in this case=26*25*24*23=358800

Case2: Password is five letters long

Similarly, the total number of password possible in this case=26*25*24*23*22=7893600

So the total number of password possible in part a) is 358800+7893600=8252400

b)

Case1: Password is four letters long

The first letter has 26 choices. The second letter has 26 choices, since the letters can be repeated. Similarly, the third letter has 26 choices, and the fourth letter has 26 choices.

Total number of password possible in this case=26^4=456976

Case2: Password is five letters long

Similarly, the total number of password possible in this case=26^5=11881376

So the total number of password possible in part b) is 456976+11881376=12338352

c)

The total number of passwords that have at least one letter used more than once is equal to the total number of passwords that may have repeated letters minus the total number of passwords in which letters cannot repeat.

Thus, he total number of passwords that have at least one letter used more than once

= 12338352 - 8252400 = 4085952

d)

Number of passwords of four letter long and all letters same=26

Number of passwords of five letter long and all letters same=26

Number of passwords of common words=71

Thus, the total number of passwords in part d)=12338352-(26+26+71)=12338229