(D) The total number of ways to select $3$ points out of $12$ is given by $^{12}C_3$. Since $5$ points are collinear,they do not form a triangle when

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

(D) The total number of ways to select $3$ points out of $12$ is given by $^{12}C_3$. Since $5$ points are collinear,they do not form a triangle when selected together. The number of ways to select $3$ points from these $5$ collinear points is $^{5}C_3$. Therefore,the total number of triangles formed is $^{12}C_3 - ^{5}C_3$. $^{12}C_3 = \frac{12 \times 11 \times 10}{3 \times 2 \times 1} = 220$. $^{5}C_3 = \frac{5 \times 4 \times 3}{3 \times 2 \times 1} = 10$. Total triangles = $220 - 10 = 210$.

Class 11 Mathematics Permutation and Combination Geometrical problems

Medium

There are $12$ points in a plane,no three of which are in the same straight line,except $5$ points which are collinear. Then the total number of triangles that can be formed with the vertices at any three of these $12$ points is

JEE MAIN 2025 All JEE MAIN

A
$230$
B
$220$
C
$200$
D
$210$

Explore More

Browse All

Question Bank

All Subjects

Class 11 Questions

Class 11

Mathematics Questions

Chapter

Permutation and Combination

Subtopic

Geometrical problems

Previous Year

JEE MAIN 2025 Paper All JEE MAIN years →

Let $p_n$ denote the total number of triangles formed by joining the vertices of an $n$-sided regular polygon. If $p_{n+1} - p_n = 66$, then the sum of all distinct prime divisors of $n$ is:

DifficultJEE MAIN 2026

View Solution

Seven white balls and three black balls are randomly placed in a row. The probability that no two black balls are placed adjacently equals

MediumAP EAMCET 2011

View Solution

If there are $m$ parallel lines in each of the two sets of parallel lines that form a parallelogram,then the total number of parallelograms formed is:

Difficult

View Solution

The number of diagonals in an octagon is

Easy

View Solution

The number of diagonals of a polygon is $35$. If $A$ and $B$ are two distinct vertices of this polygon,then the number of all those triangles formed by joining three vertices of the polygon having $AB$ as one of its sides is:

EasyTS EAMCET 2023

View Solution

Vedclass Products

For Students

Vedclass Test Series

Mock tests in real JEE/NEET style with performance analysis. 5-day free trial.

Start Free Trial

For Teachers

Exam Paper Generator

Generate Set A/B/C/D exam papers from 7.5L+ questions in 2 minutes. 3 chapters free.

Try Free

For Institutes

Online Exam Module

Live online exams with unlimited students, 360° analytics & white-label branding.

See Demo

There are $12$ points in a plane,no three of which are in the same straight line,except $5$ points which are collinear. Then the total number of triangles that can be formed with the vertices at any three of these $12$ points is

Similar Questions

Let $p_n$ denote the total number of triangles formed by joining the vertices of an $n$-sided regular polygon. If $p_{n+1} - p_n = 66$, then the sum of all distinct prime divisors of $n$ is:

Seven white balls and three black balls are randomly placed in a row. The probability that no two black balls are placed adjacently equals

If there are $m$ parallel lines in each of the two sets of parallel lines that form a parallelogram,then the total number of parallelograms formed is:

The number of diagonals in an octagon is

The number of diagonals of a polygon is $35$. If $A$ and $B$ are two distinct vertices of this polygon,then the number of all those triangles formed by joining three vertices of the polygon having $AB$ as one of its sides is:

Vedclass Test Series

Exam Paper Generator

Online Exam Module