CAT 2024 Slot 1 QA Q9 Permutations & Probability

Question 9 of the Quantitative Ability (QA) section, CAT 2024 Slot 1, with CAT's official answer and a worked solution.

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 digit natural number. Then, the value of \((a+b+c+d+n)\) is

TITA: type in the answer. There are no options and no negative marking.

Answer: 31 Permutations & Probability TITA

Solution

With 4 distinct digits there are \(4! = 24\) numbers; each digit appears \(6\) times in each of the 4 places.
Sum \(= 6 \times 1111 \times (a + b + c + d) = 6666 \times S\).
\(6666 \times 23 = 153318\), and \(153318 = 153310 + 8\), so \(S = 23\) and \(n = 8\).
\(a + b + c + d + n = 23 + 8 = 31\).

Shortcut. \(153310 / 6666 \approx 22.999\), so \(S = 23\) straight away.