TITLE: What is the value of $m+n$??
QUESTION [6 upvotes]: Suppose that there are $5$ red points and $4$ blue points on a circle . Let $\frac{m}{n}$ be the probability that a convex polygon whose vertices are among the 9 points has at least one blue vertex when $m$ & $n$ are relatively prime. Then the value of $(m+n)$ is?
This question came in recent JEE Advanced (unofficial) test and I'm stumped on how to approach this problem. The answer given is $460$ .
I'm new to geometric probability and permutational concepts.
Please someone help me out on how to approacch this problem.

REPLY [5 votes]: (# convex polygons)= (# subsets of 9-element set of size 3 or more) = 
    $512 -(36+9+1) = 466$.
(# all red polygons) = (# subsets of 5-element set of size 3 or more) = $10+5+1 = 16$.
So, $ m/n = (466 - 16) / 466 = 450/466 = 225/233 $
Thus, I get  $ m + n = 458 $  
Not sure where I miscalculated :)