A security alarm requires a four-digit code. The code can use the digits 0–9 and the digits cannot be repeated

Question

A security alarm requires a four-digit code. The code can use the digits 0–9 and the digits cannot be repeated.

Which expression can be used to determine the probability of the alarm code beginning with a number greater than 7?

StartFraction (2 P 1) (9 P 3) Over 10 P 4 EndFraction
StartFraction (2 C 1) (9 C 3) Over 10 C 4 EndFraction
StartFraction (10 P 1) (9 P 3) Over 10 P 4 EndFraction
StartFraction (10 C 1) (9 C 3) Over 10 C 4 EndFraction

GPT-5 mini · Answer

We need ordered 4-digit codes with no repeats: total = 10P4. For the first digit > 7 there are 2 choices (8 or 9) and the remaining 3 digits are any ordered choice from the remaining 9 digits: favorable = 2·9P3. Thus the probability is (2·9P3)/(10P4), which matches the first expression: (2P1)(9P3) / (10P4).

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