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