2025/26 AHSMC Part 1 Problem 5

Problem 5 of 16

The four distinct digits $1$ , $3$ , $5$ and $7$ are used to form all possible $4$ -digit numbers with no repeated digits. What is the sum of all these possible $4$ -digit numbers?

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Solution

There are $4! = 24$ such numbers, and each of $1,3,5,7$ appears equally often ( $6$ times) in each place. The digit sum is $16$ , so the total of all the numbers is $6\cdot 16\cdot(1000+100+10+1) = 106656.$ Equivalently, the average digit is $4$ , so the sum is $4444\cdot 24 = 106656$ . The answer is (C).

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