Question

How many triangles can be formed by 15 points, in which 7 of them lie on one line and the remaining 8 on another parallel line?

Answer 1

To form a triangle we need 3 non-collinear points.

Take the 7 points lying on one line be group A and the remaining 8 points lying on another parallel line be group B.

We have the following possibilities

Group A 7 points		Group B 8 points	Combination
(i)	2	1	7C2 × 8C1
(ii)	1	2	7C1 × 8C2

∴ Required number of ways of forming the triangle

= (⁷C₂ × ⁸C₁) + (⁷C₁ × ⁸C₂)

= `(7!)/(2!(7 - 2)!) xx 8 + 7 xx (8!)/(2!(8 - 2)!)`

= `(7!)/(2 xx 5!) xx 8 + 7 xx (8!)/(2! xx 6!)`

= `(7 xx 6 xx 5! xx 8)/(2! xx 5!) + (7 xx 8 xx 7 xx 6!)/(2! xx 6!)`

= `(7 xx 6 xx 8)/(2! xx 5!) + (7 xx 8 xx  xx 6!)/(2! xx 6!)`

= `(7 xx 6 xx8)/(2 xx 1) + (7 xx 8 xx 7)/(2 xx 1)`

= 7 × 6 × 4 + 7 × 4 × 7

= 168 + 196

= 364