Calculate the central angle representing men.

The sum of angles in a pie chart is $360^\circ$.
The angle for women is $120^\circ$.
The angle for children is $169^\circ$.
The angle for men is the total angle minus the sum of the angles for women and children.

$$\text{Angle for Men} = 360^\circ - (120^\circ + 169^\circ)$$ $$\text{Angle for Men} = 360^\circ - 289^\circ$$ $$\text{Angle for Men} = 71^\circ$$

The fraction of men is the angle for men divided by the total angle of the circle.

$$\text{Fraction of Men} = \frac{71^\circ}{360^\circ}$$

The total population of the city is 1,800,000.

$$\text{Number of Men} = \text{Fraction of Men} \times \text{Total Population}$$ $$\text{Number of Men} = \frac{71}{360} \times 1,800,000$$ $$\text{Number of Men} = 71 \times \frac{1,800,000}{360}$$ $$\text{Number of Men} = 71 \times 5000$$ $$\text{Number of Men} = 355,000$$

The number of men in the city is 355,000. This corresponds to option (c).

The final answer is $\text{(c) 355,000}$.