UPSC Prelims 2025 CSAT, question 58

Topic: Arrangements and puzzles

Q58

The 5-digit number $PQRST$ (all distinct digits) is such that $T \ne 0$ . $P$ is thrice $T$ . $S$ is greater than $Q$ by 4, while $Q$ is greater than $R$ by 3. How many such 5-digit numbers are possible ?

(a) 3
(b) 4Correct answer
(c) 5
(d) 6

Solution.

Answer: (b) 4

Given $T \ne 0$ and $P = 3T$ , the pair $(T, P)$ can only be $(1, 3), (2, 6),$ or $(3, 9)$ .
Given $Q = R + 3$ and $S = Q + 4 = R + 7$ , the triple $(R, Q, S)$ can only be $(0, 3, 7), (1, 4, 8),$ or $(2, 5, 9)$ .
Checking which of the $3 \times 3 = 9$ combinations have all five digits $P, Q, R, S, T$ distinct: for $(T, P) = (1, 3)$ , only $(R, Q, S) = (2, 5, 9)$ is disjoint ( $1$ number: $35291$ ); for $(2, 6)$ , both $(0, 3, 7)$ and $(1, 4, 8)$ are disjoint ( $2$ numbers: $63072, 64182$ ); for $(3, 9)$ , only $(1, 4, 8)$ is disjoint ( $1$ number: $94183$ ), giving $1 + 2 + 1 = 4$ numbers.