2024/25 AHSMC Part 1 Problem 4

Problem 4 of 16

$n$ is a positive integer with the property that $3500 \le n \le 7500$ . The digits of $n$ are $3$ , $4$ , $5$ and $7$ in some order, without repetition, and $n$ is not divisible by $5$ . How many such $n$ exist?

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Solution

There are $4! = 24$ numbers using these four digits without repetition. Of those, $3! = 6$ end with $5$ . Three others, $3457$ , $7534$ , $7543$ , are not between $3500$ and $7500$ . Therefore there are $15$ such numbers. The answer is (C).

University of Alberta

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