NinetynineCAT 2024 Slot 2 QA Q11 Permutations & Probability
Question 11 of the Quantitative Ability (QA) section, CAT 2024 Slot 2, with CAT's official answer and a worked solution.
P, Q, R and S are four towns. One can travel between P and Q along 3 direct paths, between Q and S along 4 direct paths, and between P and R along 4 direct paths. There is no direct path between P and S, while there are few direct paths between Q and R, and between R and S. One can travel from P to S either via Q, or via R, or via Q followed by R, respectively, in exactly 62 possible ways. One can also travel from Q to R either directly, or via P, or via S, in exactly 27 possible ways. Then, the number of direct paths between Q and R is
TITA: type in the answer. There are no options and no negative marking.
Answer: 7 Permutations & Probability TITA
Solution
- Let the number of direct Q-R paths be \(x\) and R-S paths be \(y\).
- P to S: via Q \(3 \times 4 = 12\), via R \(4y\), via Q then R \(3xy\). So \(12 + 4y + 3xy = 62\), i.e. \(y(4+3x) = 50\).
- Q to R: direct \(x\), via P \(3 \times 4 = 12\), via S \(4y\). So \(x + 12 + 4y = 27\), i.e. \(x = 15 - 4y\).
- Substitute: \(y(49 - 12y) = 50 \Rightarrow 12y^2 - 49y + 50 = 0 \Rightarrow y = 2\) (the other root \(\frac{25}{12}\) is not an integer).
- Then \(x = 15 - 8 = 7\).