The three-digit, positive integer answer to this question is such that the sum of the factorials of its distinct digits is also the answer to this question.

What is the answer to this question?

The answer is 145.

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Since 7! > 1000 and the answer only has three digits, you know that the only possible digits are 0-6. Also, since 6! is 720, you can't have a six because this would also imply one of the digits being greater than 6. Which leaves only 0-5. Also, since it is a 3 digit number it must be greater than 99, meaning one of the digits must be a 5, since 2! + 3! +4! < 99. This means the number is 120 + a! + b! where a and b are between 0 and 4 inclusive.

From this, one can quickly deduce that 1 and 4 are the other two digits, making the solution $\boxed{145}$ .