Which is more secure? A password with one digit is more secure, see solution.
Practice makes perfect
a
In the English alphabet there are 26 characters. The passwords can contain any of them plus one of the 10 number digits, so altogether 36 symbols can be used to build passwords.
Characters Can Be Repeated
If characters can be repeated, then any of the possible 36 characters can be at any of the 6 positions. We can use the Fundamental Counting Principle to find the number of possible passwords.
36* 36* 36* 36* 36* 36= 36^6=2 176 782 336If characters can be repeated, then there are 2 176 782 336 different passwords.
No Characters Can Be Repeated
Since the order of the characters is important even if they cannot be repeated, we can use permutations to find the answer. We need to find the permutation of the 36 characters taken 6 at a time.
_(36)P_6=1 402 410 240
Alternatively, we can also use the Fundamental Counting Principle to get the same value. Let's consider the problem.
The first character of the password can be any of the 36 characters.
The second character of the password can be any character that was not chosen as the first. This means we have 36-1=35 possibilities.
The third character of the password can be any character that was not chosen before. We have 36-2=34 possibilities.
We can keep going until we pick all 6 characters.
The Fundamental Counting Principle tells us that we need to multiply these numbers. The first number is 36 and we need 6 numbers in the product.
36* 35* 34* 33* 32* 31=1 402 410 240
If no characters can be repeated, then there are 1 402 410 240 different passwords.
b
To compare the security of the two type of passwords, we must count them first.
Only Letters that can be Repeated
If the password only contains letters that can be repeated, then any of the possible 26 letters can be at any of the 6 positions. We can use the Fundamental Counting Principle to find the number of possible passwords.
26* 26* 26* 26* 26* 26= 26^6=308 915 776If only letters are used and are allowed to be repeated, then there are 308 915 776 different passwords.
Exactly One Digit
We again use the Fundamental Counting Principle. Let's think about a process which guarantees Abby that she has a password with exactly one digit.
Abby can pick any of the 10 digits.
Abby can pick any of the 6 positions for the digit she chose.
Abby has 26 letters, and she can pick any of these to place in any of the remaining 6-1=5 positions. She can do this 26^5 many ways.
The Fundamental Counting Principle tells us that the number of correct passwords is the product of these numbers.
10* 6* 26^5=712 882 560
If the password needs to contain exactly one digit, then there are 712 882 560 different passwords.
Security
There are more passwords with exactly one digit than the ones containing only letters.
This means that it is less likely to guess a password with one digit, so it is more secure.
