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

Question

Accepted Answer

### Step 1: Define variables  
Let the number of direct paths between **Q and R** be $$x$$, and the number of direct paths between **R and S** be $$y$$.

### Step 2: Analyze the paths from P to S  
The problem states that there are **62 possible ways** to travel from **P to S**, which can happen via:  
1. **P → Q → S**: The number of paths is $$3 	imes 4 = 12$$.  
2. **P → R → S**: The number of paths is $$4 	imes y$$.  
3. **P → Q → R → S**: The number of paths is $$3 	imes x 	imes y$$.

Thus, the total number of paths is:  
$$
4y + 12 + 3xy = 62.
$$

### Step 3: Analyze the paths from Q to R  
The problem states that there are **27 possible ways** to travel from **Q to R**, which can happen via:  
1. **Q → R (direct)**: The number of paths is $$x$$.  
2. **Q → P → R**: The number of paths is $$3 	imes 4 = 12$$.  
3. **Q → S → R**: The number of paths is $$4 	imes y$$.

Thus, the total number of paths is:  
$$
x + 12 + 4y = 27.
$$

### Step 4: Solve the equations  
We now have two equations:  
1. $$4y + 12 + 3xy = 62$$  
2. $$x + 12 + 4y = 27$$

From the second equation, solve for $$x$$:  
$$
x = 27 - 12 - 4y = 15 - 4y.
$$

Substitute $$x = 15 - 4y$$ into the first equation:  
$$
4y + 12 + 3(15 - 4y)y = 62.
$$  
Simplify:  
$$
4y + 12 + (45y - 12y^2) = 62.
$$  
Combine terms:  
$$
-12y^2 + 49y + 12 = 62.
$$  
Simplify further:  
$$
-12y^2 + 49y - 50 = 0.
$$

### Step 5: Solve the quadratic equation for $$y$$  
Divide through by $$-1$$:  
$$
12y^2 - 49y + 50 = 0.
$$

Factorize or use the quadratic formula:  
$$
y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}, 	ext{ where } a = 12, b = -49, c = 50.
$$

Calculate discriminant ($$b^2 - 4ac$$):  
$$
(-49)^2 - (4)(12)(50) = 2401 - 2400 = 1.
$$

Thus, the roots are:  
$$
y = \frac{49 \pm \sqrt{1}}{24}.
$$  
Solve for $$y$$:  
$$
y = \frac{49 + 1}{24} = \frac{50}{24} = \frac{25}{12}, \quad y = \frac{49 - 1}{24} = \frac{48}{24} = y = 2.
$$

### Step 6: Find $$x$$ when $$y = 2$$  
Substitute $$y = 2$$ into $$x = 15 - 4y$$:  
$$
x = 15 - (4 	imes 2) = x = 7.
$$

### Final Answer:  
The number of direct paths between **Q and R** is **7**.