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Suppose you have a convex $n$-gon, and a convex $m$-gon, in the plane. Take the convex hull of the $n+m$ vertices. How many combinatorially distinct hulls can be obtained, where two hulls are combinatorially distinct if, with the vertices labeled $a_i$, $i=1,\ldots,n$ and $b_j$, $j=1,\ldots,m$, any two cyclicly distinct strings of labels are considered distinct? Below the hulls are $(a_1,b_1,a_2,b_2,a_3,b_3)$ and $(a_1,a_2,b_2,b_3)$.
     Two Hulls
I realize this is elementary...

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    Elementary statement $\not\to$ elementary solution. Suggestion: for small values of $m$ it might not be so hard to compute the answer for $n=1,2,3\dots$. Maybe there will be a pattern, or maybe there will be something in the Online Encyclopedia of Integer Sequences.2012-02-03
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    @Gerry: Great suggestion to compute and compare to OEIS!!2012-02-04

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