A security keypad has 12 alpha-numeric characters on it. (a) How many different

Question

A security keypad has 12 alpha-numeric characters on it. (a) How many different four-digit security codes are possible if the repetition of values is allowed? (b) How many different four-digit security codes are possible if the values must all be different? (c) If the four values are randomly selected (with replacement) to form a security code, what is the probability that at least one value repeats in the code? (d) A security expert believes that the four digit code allowing for repetition is not secure enough. She recommends that the code consists of k values, where k is chosen such that the probability of randomly guessing the code is less than 10^-8. What is the smallest value of k for which this is true?

Explanation / Answer

A) Number of 4 digit codes possible = 124 = 20736

B) if all values must be different, number of codes possible = 12x11x 10x9 = 11880

C) P(at least 1 value repeats) = 1 - P(all values are distinct)

= 1 - 11880/20736

= 0.4271

D) here, if probability of guessing must be less than 10-8, the number of codes possible shud be 108

12k = 108

k log 12 = 8log 10

k = 7.41

So, the smallest value of k must be 8

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