Explanation:

- Initially, 150 soldiers have food lasting for 400 days.

- After 50 days, they've used 50 days' worth of food, leaving food for 350 days for 150 soldiers.

- With the original group, food left = 350 days × 150 soldiers = 52,500 soldier-days of food.

- 100 more soldiers join, making a total of 250 soldiers.

- Total food can now last for 52,500 soldier-days / 250 soldiers = 210 days.

- Thus, from day 51, the food will last for 210 days with 250 soldiers.

210 days is the correct answer.