In how many ways can $$b$$ blue balls and $$r$$ red balls be distributed in $$n$$ distinct boxes?

The exponent of $$11$$ in the prime factorization of $$300!$$ is

Let $${x_n}$$ denote the number of binary strings of length $$n$$ that contain no consecutive $$0s$$.

The value of $${x_5}$$ is

Suppose that a robot is placed on the Cartesian plane. At each step it is allowed to move either one unit up or one unit right, i.e., if it is at $$(i, j)$$ then it can move to either $$(i+1, j)$$ or $$(i, j+1)$$

How many distinct path are there for the robot to reach the point $$(10, 10)$$ starting from the initial position $$(0, 0)$$?