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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

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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

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