What is the value of $m+n$??

(# 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 :)


Assuming that a convex plolygon has at least three vertices the overall number of the convex polygons is

$$2^9-C_2^9-C_1^9-1=466,$$ where $C^i_j$ are the binomial coefficients.

From those

$$2^5-C_2^5-C^5_1-1=16$$

have only red vertices.

Thus the probability $m/n$ is $(466-16)/466=450/466=225/233$ and $m+n=458$.

Tags:

Probability

Permutations

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