  We propose a discrete approach to solve problems on forming polygons from
broken sticks, which is akin to counting polygons with sides of integer length
subject to certain Diophantine inequalities. Namely, we use MacMahon's
Partition Analysis to obtain a generating function for the size of the set of
segments of a broken stick subject to these inequalities. In particular, we use
this approach to show that for $n\geq k\geq 3$, the probability that a $k$-gon
cannot be formed from a stick broken into $n$ parts is given by $n!$ over a
product of linear combinations of partial sums of generalized Fibonacci
numbers, a problem which proved to be very hard to generalize in the past.
