A security keypad has 15 alpha-numeric characters on it. (a) How many different

Question

A security keypad has 15 alpha-numeric characters on it.

(a) How many different three-digit security codes are possible if the repetition of values is allowed?

(b) How many different three-digit security codes are possible if the values must all be different?

(c) If the three 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 three 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 108 . What is the smallest value of k for which this is true?

Thank you in advance!

Explanation / Answer

a) total 3 digit codes =N(for each digit there are 15 choices) =15*15*15 =3375

b) number of 3 digit codes =N(for first digit 15; for second 14 and for third 13 choices) =15*14*13 =2730

c) number with repetiotions =N(with repeatitions)-N(without repetitions) =3375-2730=645

hence probability=645/3375

=0.1911

d)

here probability of radomely guessing =1/15k <10-8

taking log on both sides:

k>8/log(15)

k>6.8

k=7

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