66First, we note that there are d= nC2 - n diagonals of an n sided regular polygon and two can be selected in dC2 ways.
If the two two selected diagonals intersect within the polygon, then the end points of these diagonals must form a quadrilateral. Or taking the reverse route, we see that the number of pair of intersecting diagonals is equal to the number of quadrilaterals formed by the vertices of the polygon. This number is nC4. Hence, the required probability is
p = nC4dC2
Intersetingly, if one wants to find the number, I(n) of intersections of the diagonals inside an n sided regular polygon, the result is frightening [1]:
For n ≥ 3,
I(n) =
nC4 + -5n3+45n2-70n+2424 δ2(n) - 3n2 δ4(n) + -45n2 + 262n6 δ6(n) + 42n δ12(n) + 60n δ18(n) + 35n δ24(n) -38n δ30(n) -82n δ42(n) - 330n δ60(n) -144n δ84(n) -96n δ90(n) -144n δ120(n) -96n δ210(n)
where the notation δm(n) is defined as
δm(n) = 1 if n ≡ 0 (mod m)
= 0 otherwise
Obviously, I didn't derive it. However, I can derive it (but who cares.)
62I guess or i am sure this is more frightening than any other expression ever put on this forum :D
:)
24well answer given in book is not that complex..
\frac{^nC_4}{^{^{n}C_2-n}C_2}
i didnt get this eiter..
plz help sirs..
62
Lokesh Verma
·2009-09-15 07:29:55
if you remeber the question with n sided convex polygon, then the number of points of intersection of diagonals within the polygon is given by nC4
while the number of diagonals is given by nC2-n
Now any 2 diagonals will give a point of intersection... hence that will be given by
nC2-nC2
Hence the answer :)
24
eureka123
·2009-09-15 07:32:17
so is anant sir's soln wrong ?
62
Lokesh Verma
·2009-09-15 07:37:00
The only difference between Anant Sir's answer and the one you have given is that there is no difference :D :D
Read only upto the first 2-3 lines :P
62
Lokesh Verma
·2009-09-15 07:50:15
See what anant sir has done is this :
"regular polygon" The motivation for that is that in aregular polygon, you will have a lot of points that will intersect at the same point.. example in a regular hexagon,, the main diagonals will all intersect at one single point... so we will lose 2 points due to repetition...
So the result is the general result for the case of a regular polygon
