1. Total number of ways to choose 5 numbers from 18

Since 5 distinct numbers are selected from $1,2,3,\dots,18$ and then arranged in increasing order automatically, each selection corresponds to exactly one ordered tuple $x_1<x_2<x_3<x_4<x_5.$

Hence total number of possible outcomes is $\binom{18}{5}.$

2. Condition $x_2=7$ and $x_4=11$

We need $x_1 < x_2=7 < x_3 < x_4=11 < x_5.$

Now count choices for $x_1,x_3,x_5$.

• Since $x_2=7$, we need exactly one number less than 7 before it. Thus $x_1 \in \{1,2,3,4,5,6\},$ giving $6$ choices.

• Since $x_4=11$, we need exactly one number between 7 and 11 to be $x_3$. Thus $x_3 \in \{8,9,10\},$ giving $3$ choices.

• Finally, $x_5>11$, so $x_5 \in \{12,13,14,15,16,17,18\},$ giving $7$ choices.

Therefore favorable outcomes are $6\times 3\times 7=126.$

3. Required probability

$P=\frac{126}{\binom{18}{5}}.$

Now, $\binom{18}{5}=\frac{18\cdot17\cdot16\cdot15\cdot14}{5\cdot4\cdot3\cdot2\cdot1}=8568.$

So, $P=\frac{126}{8568}=\frac{1}{68}.$

4. Option check

Thus the correct option is $\boxed{\text{C } \frac{1}{68}}.$

5. Comparison with stored answer

Stored correct answer: C

Our derived answer is also C, so they agree.