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 between RandS.OnecantravelfromPtoSeitherviaQ,orviaR,orviaQfollowed 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:

The correct option is 7.

To find the number of direct paths between towns Q and R, we can define the variables and represent the connections between the towns mathematically. Let:
- $N_{\mathrm{PQ}}$ be the number of direct paths between P and Q, so $N_{\mathrm{PQ}}=3$.
- $N_{\mathrm{QS}}$ be the number of direct paths between Q and S, so $N_{\mathrm{QS}}=4$.
- $N_{\mathrm{PR}}$ be the number of direct paths between P and R, so $N_{\mathrm{PR}}=4$.
- $N_{\mathrm{PS}}$ be the number of direct paths between P and S, so $N_{\mathrm{PS}}=0$.
- $x$ be the number of direct paths between Q and R.
- $y$ be the number of direct paths between R and S.

Step 1: Formulate the first equation using paths from P to S
According to the problem, one can travel from P to S either via Q, or via R, or via Q followed by R, in exactly 62 possible ways. We can express the number of ways for each route as follows:
1. Path via Q (P → Q → S):
$N_{\mathrm{PQ}}\times N_{\mathrm{QS}}=3\times 4=12$ ways.
2. Path via R (P → R → S):
$N_{\mathrm{PR}}\times N_{\mathrm{RS}}=4\times y=4y$ ways.
3. Path via Q followed by R (P → Q → R → S):
$N_{\mathrm{PQ}}\times N_{\mathrm{QR}}\times N_{\mathrm{RS}}=3\times x\times y=3xy$ ways.

Summing these options gives the total number of ways to travel from P to S:
$12+4y+3xy=62$
Subtracting 12 from both sides, we get:
$y(4+3x)=50$  — (Equation 1)

Step 2: Formulate the second equation using paths from Q to R
The problem states that one can travel from Q to R either directly, or via P, or via S, in exactly 27 possible ways:
1. Directly (Q → R):
$x$ ways.
2. Via P (Q → P → R):
$N_{\mathrm{QP}}\times N_{\mathrm{PR}}=3\times 4=12$ ways.
3. Via S (Q → S → R):
$N_{\mathrm{QS}}\times N_{\mathrm{SR}}=4\times y=4y$ ways.

Summing these options gives the total number of ways to travel from Q to R:
$x+12+4y=27$
Subtracting 12 from both sides, we get:
$x+4y=15$
From this, we can express $y$ in terms of $x$:
$y=\frac{15-x}{4}$  — (Equation 2)

Step 3: Solve the system of equations
Substitute Equation 2 into Equation 1:
$\left(\frac{15-x}{4}\right)(4+3x)=50$
Multiply both sides by 4:
$(15-x)(4+3x)=200$
Expand the terms:
$60+45x-4x-3x^2=200$
Simplify and rearrange the equation into a standard quadratic form:
$3x^2-41x+140=0$

Solve the quadratic equation by factoring:
$3x^2-21x-20x+140=0$
$3x(x-7)-20(x-7)=0$
$(3x-20)(x-7)=0$

This yields two potential solutions:
$x=7$   or   $x=\frac{20}{3}$

Since the number of paths between towns must be a non-negative integer, we discard the fractional value. Thus:
$x=7$

Substituting $x=7$ back into Equation 2 confirms that $y=\frac{15-7}{4}=2$, which is also a valid integer.

Therefore, the number of direct paths between Q and R is 7.