Use app×
Join Bloom Tuition
One on One Online Tuition
JEE MAIN 2025 Foundation Course
NEET 2025 Foundation Course
CLASS 12 FOUNDATION COURSE
CLASS 10 FOUNDATION COURSE
CLASS 9 FOUNDATION COURSE
CLASS 8 FOUNDATION COURSE
0 votes
74 views
in Mathematics by (114k points)
closed by
Out of 15 points in plane, n points are in the same straight line, 445 triangles can be formed by joining these points. What is the value of n?
1. 3
2. 4
3. 5
4. 6
5. None of these

1 Answer

0 votes
by (115k points)
selected by
 
Best answer
Correct Answer - Option 3 : 5

Concept:

Number of ways to select 3 points out of the n collinear points = \({\;^n}{C_3}\)

\({\;^n}{C_r}\; = \;\frac{{n!}}{{r!\left( {n\; - \;r} \right)!}}\)

 

Calculation:

Number of triangles that can be formed is equal to the number of ways to select 3 non-collinear points.

⇒ Number of ways to select 3 points from 15 points = 15c3

Let n points be collinear.

⇒ Number of ways to select 3 points out of the n collinear points = nc3

So, Number of ways to select 3 non-collinear points = (Number of ways to select 3 points using all the points - Number of ways to select 3 points using the collinear points)

⇒ Number of ways to select 3 non-collinear points = 15c3 - nc3

⇒ Number of triangles that can be formed = 15c3 - nc3

⇒ 445 = 15c3 - nc3

nc3 = 15c3 – 445 = 455 – 445 = 10

\(\Rightarrow \frac{{n!}}{{\left( {n - 3} \right)!\; \times 3!}} = 10\)

\(\Rightarrow \frac{{n\left( {n - 1} \right)\left( {n - 2} \right)}}{6} = 10\)

⇒ n (n – 1) (n – 2) = 60

∴ n = 5

Welcome to Sarthaks eConnect: A unique platform where students can interact with teachers/experts/students to get solutions to their queries. Students (upto class 10+2) preparing for All Government Exams, CBSE Board Exam, ICSE Board Exam, State Board Exam, JEE (Mains+Advance) and NEET can ask questions from any subject and get quick answers by subject teachers/ experts/mentors/students.

Categories

...