Answer: let n be the number of points in a plane k be the number of point to form a triangle Step-by-step explanation: combination of numbers C (n, k) = n! / k(n -k)! = 15! / 3(15 - 3)! = 15! / 3(12)! = (15x14x13) / (3x2) = 2730 / 6 =455 thus there are 455 triangles 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?This question was previously asked in NDA (Held On: 18 Sept 2016) Maths Previous Year paper View all NDA Papers >
Answer (Detailed Solution Below)Option 3 : 5 Free Electric charges and coulomb's law (Basic) 10 Questions 10 Marks 10 Mins 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 Last updated on Sep 29, 2022 Union Public Service Commission (UPSC) has released the NDA Result II 2022 (Name Wise List) for the exam that was held on 4th September 2022. Earlier, the roll number wise list was released by the board. A total number of 400 vacancies will be filled for the UPSC NDA II 2022 exam. The selection process for the exam includes a Written Exam and SSB Interview. Candidates who get successful selection under UPSC NDA II will get a salary range between Rs. 15,600 to Rs. 39,100.
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How many triangles can be formed by joining 10 points on a plane in which no three points are collinear?110 triangles can be formed by joining 10 points as vertices, in which n points are collinear.
How many triangles are determined by 12 points no 3 points of which are collinear?Solution : From the 'n' points (no three points are collinear) in a plane, the number of triangle formed is `. ^(n)C_(3)`. <br> `therefore` The required number of triangles <br> `=. ^(12)C_(3)=(12xx11xx10)/(1xx2xx3)=220`.
How many triangles can be formed by joining 15 points when 7 of them are on the same straight line?There will be 7C2 = 21 possible combinations for the two vertices on the line with 7 points. 8 x 21 = 168 triangles.
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