Math, asked by faithyetty1431, 1 year ago

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

Answers

Answered by sprao534
4

Please see the attachment

Attachments:
Answered by mad210218
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

Similar questions