Question

Find the number of triangles that can be formed using 14 points in a plane such that 4 points are collinear?

Answer 1

Please see the attachment

Answer 2

Given :

14 points in a plane

4 points are collinear.

To find :

Numbers of triangles made by these points

Solution :

There are total 14 points

to make a triangle we have to choose three points from all 14,

but

it is given that 4 points are collinear

So in these four points,we cant choose any three points at once.

To choose r points out of n without any condition, the formula of combination is used as

$\bf \binom{n}{r} = ^{n} C _{r} = \frac{n!}{r!(n - r)!} \:$

(equation 1)

If we have n points in a plane in which m points are collinear then number of triangles formed by those n points are :

$^{n} C _{3} - ^{m} C _{3}$

(equation 2)

By using equation 1 in equation 2.

The number of triangle formed :

$\bf ^{14} C _{3} - ^{4} C _{3} -= \frac{14!}{3!(14 - 3)!} - \frac{4!}{3!(4 - 3)!} \\ \\ = \frac{14!}{3!(11)!} - \frac{4!}{3!(1)!}$

So,

number of triangles formed :

$= \frac{14 \times 13 \times 12}{3 \times 2 \times 1} - \frac{4}{1}$

$= (14 \times 13 \times 2) - 4$

So the number of triangles formed = 364 -4 = 360