Circle Problem From 3Blue1Brown （分圆问题）

$\text{Background}$

$\text{Last night, lots of students from primary school came to our class to study OI.}$
$\text{Mark next to me was asked by one of them, "Dude, are you copying codes?"}$
$\text{Mark was very angry that time cuz he's just moving his code from IDE to blog.}$
$\text{So he decided to let them take a HARD test (though he failed).}$
$\text{Finally that guy apologized and we started upgrading our test so that it could }$$\text{be harder. They'll take this test this Sunday.}$

$\text{Problem}$

$\text{Interger }N\text{ and }N\text{ points in a circle are given. Connect every pair of }$
$\text{these points to a edge. There aren't any 3 edges which shares one point.}$
$\text{Please calculate how many pieces of the circle are cut by these edges.}$

$\text{Solution}$

$\text{Let's consider some cases with smaller }N\text{s.}$
$\text{Easy to get}$

$N$	$ans$
$1$	$1$
$2$	$2$
$3$	$4$
$4$	$8$
$5$	$16$
$...$	$...$

$\text{Dude, }ans=2^{N-1}.\text{ Solved.}$


$\text{But actually it's wrong.}$
$\text{These formula }$JUST$\text{ right when }N\in\{1,2,3,4,5,6,10\}.$

$\text{Let's do some simple problems first.}$

$\text{I. Calculate how many edges are there in the circle;}$

$\because\text{Every 2 points make a edge, and there're }N\text{ points,}$
$\therefore\text{There're }C_{N}^{2}\text{ edges in total.}$

$\text{II. Calculate how many points of intersection of these edges.}$

$\text{It maybe a little difficult, but I think it's necessary for you guys to think about it.}$

$\because\text{Every 2 edges make a point of intersection, every 2 points make a edge,}$
$\text{and there're }N\text{ points,}$
$\therefore\text{There're }$$N\times(N-1)\times(N-2)\times(N-3)$$=C_{N}^{4}\text{ points of intersection in total.}$


$\text{Here goes our Euler's formula in topology. Set a polyhedron which has }$
$V\text{ vertexes ,}F\text{ pieces of surface, and }E\text{ edges. It always meet}$$V-E+F=2$
$\therefore F=E-V+2.$

$\text{Therefore, the answer is }C_{N}^{2}+C_{N}+{4}+2-1=C_{N}^{2}+C_{N}+{4}+1.\text{ Solved.}$

$\text{The End}$

$\text{Video by 3Blue1Brown from bilibili}$
$\text{Reference material}$