A certain application requires power at a frequency of 16.67 Hz, while the available grid frequency is 50 Hz. A 3-phase synchronous motor connected to the 50 Hz, 3-phase grid, driving a synchronous generator, is used for this application. Which one of the following combinations is a suitable choice for the number of poles in the motor and generator, respectively?

The correct option is 6, 2.

To understand why this is the correct choice, let's look at the relationship between frequency, speed, and the number of poles in synchronous machines.

A synchronous motor and a synchronous generator are mounted on the same shaft, which means they must rotate at the exact same mechanical speed.

The synchronous speed of an electrical machine, denoted as $N_s$ in revolutions per minute (rpm), is given by the formula:
$N_s=\frac{120\times f}{P}$
where:
- $f$ is the frequency of the electrical power (in Hz)
- $P$ is the number of poles of the machine

Let $P_m$ be the number of poles of the motor connected to the grid frequency $f_m=50\text{Hz}$.
Let $P_g$ be the number of poles of the generator producing the application frequency $f_g=16.67\text{Hz}$ (which is approximately $\frac{50}{3}\text{Hz}$).

Since both machines are mechanically coupled and run at the same speed ($N_s$), we can equate their speeds:
$\frac{120\times f_m}{P_m}=\frac{120\times f_g}{P_g}$

Simplifying the equation by dividing both sides by 120 gives:
$\frac{f_m}{P_m}=\frac{f_g}{P_g}$

Rearranging this to find the ratio of the number of poles of the motor to the number of poles of the generator:
$\frac{P_m}{P_g}=\frac{f_m}{f_g}$

Substituting the given frequencies ($f_m=50\text{Hz}$ and $f_g=16.67\text{Hz}$):
$\frac{P_m}{P_g}=\frac{50}{16.67}\approx 3$

This means the motor must have three times as many poles as the generator:
$P_m=3\times P_g$

Now let's check the given options to see which one satisfies this ratio, keeping in mind that the number of poles must be an even integer:
- For motor poles $P_m=6$ and generator poles $P_g=2$:
$\frac{6}{2}=3$
This perfectly matches the required ratio.

Therefore, a suitable choice for the number of poles in the motor and generator is 6 and 2, respectively.