# There are 12 distinct points A, B, C, …, L, in order, on a circle. Lines are drawn passing through each pair of points.

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There are 12 distinct points A, B, C, …, L, in order, on a circle. Lines are drawn passing through each pair of points.

(i) How many lines are there in total?

(ii) How many lines pass through D?

(iii) How many triangles are determined by lines?

(iv) How many triangles have on vertex C?

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(i) We need two points to draw a line.

∴ Total number of lines = 12C2 = 66

(ii) Lines are drawn passing through each pair of points.

∴ Lines from point D will pass through all the remaining 11 points.

∴ 11 lines pass through D.

(iii) We need three points to draw a triangle.

∴ Number of triangles = 12C3 = 220

(iv) To get the triangles with one vertex as C, we need two vertices from the remaining 11 vertices

∴ Number of triangles with vertex at C = 11C2

$\frac{11\times10}2$ = 55