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permutation & combination

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if no 3 diagonal of a convex decagon meet at the same point inside the decagon ,into how many line segments are the diagonal divided by their intersection?
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Since for every 4 vertices we can count exactly 1 intersection between the diagonals so total number of intersections would be C(10 , 4) = 210 intersections between the diagonals ,

and also the number of diagonals would be C(10 , 2) -10 = 45 - 10 = 35 as there are C(10 , 2) straight lines joining the C(10 , 2) pairs of vertices ,but 10 of these 45 lines are the sides of the decagon.

The number of line segments are k+1 when there are k intersections along a line and each intersecting point lies on two diagonals.

we start with 35 diagonals. Each intersection point adds a segment to both of the intersecting diagonals.

Therefore the total number of straight line segments into which the diagonals are divided is

answer is 35 + twice number of intersections.

 = 35+2×210.

= 455 Ans

For detailed explanation please refer here : http://vle.du.ac.in/mod/book/print.php?id=5284&chapterid=402

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