A box contains $6$ identical red balls and $5$ identical blue balls. How many distinct ways can these balls be arranged in a row?

The total number of objects to be arranged is:
$$n=6+5=11\text{ balls}$$

Out of these, $6$ are of one identical kind (red) and $5$ are of another identical kind (blue).
The number of unique horizontal permutations is given by the formula:

$$P=\frac{n!}{p!\cdot q!}=\frac{11!}{6!\cdot 5!}$$

$$P=\frac{11\times 10\times 9\times 8\times 7\times 6!}{6!\times (5\times 4\times 3\times 2\times 1)}$$

$$P=\frac{11\times 10\times 9\times 8\times 7}{120}$$

$$P=11\times 3\times 2\times 7=462$$

Therefore, there are $462$ distinct ways to arrange the balls.