CAT 2025 Slot 3 QA Question 14: The sum of all four-digit numbers that can be formed with the distinct non-zero digits $a, b, c$ and $d$, with each digit appearing exactly once in every number, is $153310 + n$, where $n$ is a single

Question

The sum of all four-digit numbers that can be formed with the distinct non-zero digits $a, b, c$ and $d$, with each digit appearing exactly once in every number, is $153310 + n$, where $n$ is a single

Accepted Answer

Let $S = a + b + c + d$.
There are $4! = 24$ four-digit numbers. In these $24$ numbers, each digit appears $6$ times in each place.
Sum of all numbers = $6S(1000 + 100 + 10 + 1)$
$= 6666S$
We need $153310 + n$ to be a multiple of $6666$, where $n$ is a single digit natural number.
$6666	imes 23 = 1