A factory produces $120$ toys in $8$ days. How many toys will it produce in $15$ days at the same production rate?

Since the rate of production remains constant, the number of toys produced is directly proportional to the number of days.
1. Find the rate of production per day:

$$\text{Production per day}=\frac{\text{Total toys produced}}{\text{Total days}}=\frac{120}{8}=15\text{ toys/day}$$

2. Calculate the total production for $15$ days:

$$\text{Production in 15 days}=\text{Production per day}\times 15\text{ days}$$

$$\text{Production in 15 days}=15\times 15=225\text{ toys}$$

Therefore, the factory will produce $225$ toys in $15$ days.