1. Initial Position:

• A starts at his house and walks $20$ m East.

2. B Joins A:

• B joins A after A walks $20$ m East, and together they walk $10$ m East.
• Now, they are both $30$ m East from A's starting point.

3. Turns:

• A turns left (to the North) and walks $2$ m.
• B turns right (to the South) and walks $8$ m.

4. Further Movements:

• B turns left (to the East) and walks $4$ m.
• B then turns right (to the South) and walks $5$ m to reach his office.
• A turns right (to the East) and walks $12$ m to reach his office.

5. Final Positions:

• A’s final position: A is now $30$ m East + $2$ m North + $12$ m East = $42$ m East, $2$ m North.
• B’s final position: B is $30$ m East + $8$ m South + $4$ m East + $5$ m South = $34$ m East, $13$ m South.

6. Finding the Shortest Distance Between the Two Offices:

• A’s office is at ($42$ m East, $2$ m North).
• B’s office is at ($34$ m East, $13$ m South).

7. Shortest Distance Calculation: Distance = $\sqrt{[(42 - 34)^2 + (2 - (-13))^2]}$ Distance = $\sqrt{[8^2 + (2 + 13)^2]} = \sqrt{[64 + 225]} = \sqrt{289} = 17$ m

Answer: B. $17$ m