A three-digit PIN is randomly selected using the digits 0–9. The digits can be repeated. What is the probability that there are no repeated digits?

Rationale

The probability that there are no repeated digits in a three-digit PIN selected from the digits 0–9 is 0.72.

To determine this probability, we need to consider the total number of possible three-digit combinations and the number of combinations that have no repeating digits. The total number of combinations is 10^3, while the combinations without repetition can be calculated as 10 × 9 × 8, yielding a probability of 0.72.

A (0.72) CORRECT

This choice correctly reflects the probability of forming a three-digit PIN with no repeated digits. With 10 choices for the first digit, 9 for the second, and 8 for the third, the calculation yields a total of 720 successful combinations. Dividing this by the 1000 total combinations gives 0.72.

B (0.7) YOUR ANSWER

This choice underestimates the probability of selecting a three-digit PIN without repeated digits. The correct calculation shows that the probability is actually higher than 0.7, specifically 0.72, as demonstrated through the combination formulas.

C (0.3) YOUR ANSWER

This value is significantly lower than the correct probability. The number 0.3 does not reflect the combinatorial count of non-repeating digit selections, which is much greater than would be suggested by this option.

D (1) YOUR ANSWER

This choice incorrectly suggests that every possible three-digit PIN will have no repeated digits, which is not true. Since digits can repeat, the maximum probability cannot be 1, and thus this option is invalid.

Conclusion

The probability of selecting a three-digit PIN with no repeated digits is calculated to be 0.72, based on the combinatorial options available. Understanding the total and successful outcomes reveals that while repetition is allowed, the calculation of probabilities must accurately reflect the constraints imposed by the lack of repeats. Hence, only option A correctly represents this scenario.

A three-digit PIN is randomly selected using the digits 0–9. The digits can be repeated. What is the probability that there are no repeated digits?

The probability that there are no repeated digits in a three-digit PIN selected from the digits 0–9 is 0.72.

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