Question

Kiara sets up a passcode on her tablet, which allows only 7 digit codes. she has heard that it’s more secure if a digit is repeated in the passcode. A spy sneaks a look at Kiara’s tablet and sees her fingerprints on the screen over 6 numbers. what is the probability the spy is able to unlock the tablet on his first try? express your answer as a fraction

Answer 1

Lets suppose that the numbers that the spy saw were 1,2,3,4,5,6 and by the question we know that the password has 7 digits, meaning that one of those number repets itself. We dont know wich one repets, so we need to keep that in mind. First, lets imagine that the number 1 repeats, so one of the possibilities is:

$1123456$

We can see that to find all the possibilities of passwords with 1 repeating is a permutation with repetition. So the amount of passwords that we can have is:

$\text{Passwords with 1 repeating=}\frac{7!}{2!}=7\times6\times5\times4\times3=2520$

Now, we just need to calculate the possibilities when 2,3,4,5 or 6 repeats:

$\text{Passwords with 2 repeating=}\frac{7!}{2!}=7\times6\times5\times4\times3=2520$$\text{Passwords with 3 repeating=}\frac{7!}{2!}=7\times6\times5\times4\times3=2520$$\text{Passwords with 4 repeating=}\frac{7!}{2!}=7\times6\times5\times4\times3=2520$$\text{Passwords with 5 repeating=}\frac{7!}{2!}=7\times6\times5\times4\times3=2520$$\text{Passwords with 6 repeating=}\frac{7!}{2!}=7\times6\times5\times4\times3=2520$

So, in total we have 15120 possibilities of passwords, so the probability of a spy getting the password first try is:

$probability\text{ first try=}\frac{1}{15120}$