The author claims that it is safer to have a repeated digit

The author claims that it is safer to have a repeated digit in your 4 digit PIN than not. Since the number of 4 digit PINs is exactly 10,000, and a majority of them (5040) have no repeated digit. Is there any validity to this author\'s claim? How does this claim play out for 5 digit PINs and 6 digit PINs?

Solution

Since the pin contains 4 number

The first number of pin can be choosen from 10 possible combinations i.e. 0 to 9

The second number can be choosen in 9 ways to avoid repetition

The third number can be choosen in 8 ways to avoid repetition

The fourth number can be choosen in 7 ways to avoid repetition

Number of ways with no digit repeated = 10 * 9 * 8 * 7 = 5040 ways

Yes, the statement that we have 5040 possibilites with no digit repeated is true

Claim for 5 digit Pin = 10 * 9 * 8 * 7 * 6 = 30240 ways

So with repetititon we have 100000 - 30240 = 69760 ways (hence it is better to have repetition in the case of 5 digit pin)

Claim for 5 digit Pin = 10 * 9 * 8 * 7 * 6 * 5 = 151200 ways

So with repetititon we have 1000000 - 151200= 848800 ways (hence it is better to have repetition in the case of 6 digit pin)