In how many ways can n red balls and n blue balls be arranged such that number of red balls from left side is >= number of blue balls?

Question

In how many ways can 'n' red and 'n' blue balls be arranged, such that at any point from the left hand side, the number of red balls is always greater than or equal to the number of blue balls ? Example : n = 2 RRBB RBRB Answer = 2

Answer 1

This is exactly equal to the $n$-th Catalan number. They're a well studied and interesting group of numbers, and arise from a fundamental recurrence relation which you can actually obtain from your problem as well. This is $$C_n=\sum_{k=0}^{n-1}C_kC_{n-k-1}$$, where $C_0=1$. They also have a closed form binary expression, and you can find a derivation starting from the recurrence here.

Another useful link is this. You will notice that the same problem can be reformulated in a number of ways, such as the number of acceptable parantheses sequences (which is exactly your problem, replace red with '(' and blue with ')'), monotonic lattice paths, and what are known as Dyck paths (essentially the same).