(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Suppose that [tex] n [/tex] points lie on a circle are all joined in pairs. The points are positioned so that no three joining lines are concurrent in the interior of the circle. Let [tex] a_n [/tex] be the number of regions into which the interior of the circle is divided. Draw diagrams to find [tex] a_n [/tex] is given by the following formula [tex] a_n = n + \binom{n-1}{2} + \binom{n-1}{3} + \binom{n-1}{4} = 1+ \frac{n(n-1)(n^{2}-5n+18)}{24} [/tex].

2. Relevant equations

Maybe use strong induction or [tex] P(n \leq k) \Rightarrow P(k+1) [/tex]

3. The attempt at a solution

Here is what I drew: http://uploadimages.com/view.php?type=thumb3&p=2007/0081/11820544788596.jpg" [Broken]

I think strong induction is just a special case of regular induction. So really we can always use strong induction. I cant seem to find a pattern.

There seems to be a pattern for [tex] a_n = 1,2,3 \ldots 6 [/tex] (i.e. [tex] 1, 2, 4, 8, 16, 31, 57, 99, 163 [/tex]). But then the "powers of two" pattern ends after [tex] n = 5 [/tex]. So I am not sure strong induction could be used.

But how would I prove the inductive step? Would I use the definition of the binomial coefficient?

Thanks

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# Homework Help: Circle Proof

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