Question

a 6 - digit pin number is selected. what it the probability that there are no repeated digits? the probability that no numbers are repeated is. write your answer in decimal form, rounded to the nearest thousandth.

Explanation:

Step1: Calculate total number of 6 - digit PINs

Each digit of the PIN can be chosen in 10 ways (digits 0 - 9). So the total number of 6 - digit PINs is \(N = 10\times10\times10\times10\times10\times10=10^{6}\)

Step2: Calculate number of 6 - digit PINs with no repeated digits

For the first digit, we have 10 choices. For the second digit, we have 9 choices (to avoid repetition). For the third digit, we have 8 choices, for the fourth digit 7 choices, for the fifth digit 6 choices and for the sixth digit 5 choices.
The number of 6 - digit PINs with no repeated digits is \(M=P(10,6)=\frac{10!}{(10 - 6)!}=\frac{10!}{4!}=10\times9\times8\times7\times6\times5 = 151200\)

Step3: Calculate the probability

The probability \(P\) that there are no repeated digits is \(P=\frac{M}{N}=\frac{151200}{1000000}=0.1512\)

Answer:

\(0.151\)