Question

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 a few direct paths between Q and R and also 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:

• A. 0
• B. 2
• C. 3
• D. 4
• E. 5

Hint:

In path counting problems, use multiplication for sequential paths and addition for alternative routes. Set up equations based on the given totals.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
The number of ways to travel from A to C via B is the product of paths between (A, B) and (B, C). Total ways is the sum of paths of independent routes.
Step 2: Key Formula or Approach:
Let \( x = \) direct paths between Q and R, and \( y = \) direct paths between R and S.
Total paths \( P \to S = (P \to Q \to S) + (P \to R \to S) + (P \to Q \to R \to S) \).
Total paths \( Q \to R = (Q \to R) + (Q \to P \to R) + (Q \to S \to R) \).
Step 3: Detailed Explanation:
Given: \( PQ = 3, QS = 4, PR = 4 \).
1. Condition 1 (P to S):
\[ (3 \times 4) + (4 \times y) + (3 \times x \times y) = 62 \]
\[ 12 + 4y + 3xy = 62 \implies 4y + 3xy = 50 \dots(i) \]
2. Condition 2 (Q to R):
\[ x + (3 \times 4) + (4 \times y) = 27 \]
\[ x + 12 + 4y = 27 \implies x + 4y = 15 \implies 4y = 15 - x \dots(ii) \]
Substitute (ii) into (i):
\[ (15 - x) + 3x \times \frac{15 - x}{4} = 50 \]
\[ 60 - 4x + 45x - 3x^2 = 200 \]
\[ 3x^2 - 41x + 140 = 0 \]
Solving the quadratic equation: \( (x - 7)(3x - 20) = 0 \).
Since \( x \) must be an integer, \( x = 7 \).
Step 4: Final Answer:
The number of direct paths between Q and R is 7.