Question:
There are \(10\) points in a plane, out of these \(6\) are collinear. If \(N\) is the total number of triangles formed by joining these points, then
\[
N=
\]
There are \(10\) points in a plane, out of these \(6\) are collinear. If \(N\) is the total number of triangles formed by joining these points, then
\[
N=
\]
Show Hint
When some points are collinear, first count all possible selections of \(3\) points and then subtract the selections made entirely from the collinear points.
Updated On: Jun 22, 2026
- \(120\)
- \(850\)
- \(100\)
- \(150\)
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The Correct Option is C
Solution and Explanation
Step 1: Find total ways to choose \(3\) points from \(10\) points.
A triangle is formed by selecting \(3\) non-collinear points.
Total ways to select \(3\) points from \(10\) points: \[ {}^{10}C_3 \] \[ {}^{10}C_3=\frac{10\cdot 9\cdot 8}{3\cdot 2\cdot 1} \] \[ =120 \]
Step 2: Subtract the invalid selections.
Since \(6\) points are collinear, choosing any \(3\) points from these \(6\) points will not form a triangle.
Number of invalid selections: \[ {}^6C_3 \] \[ {}^6C_3=\frac{6\cdot 5\cdot 4}{3\cdot 2\cdot 1} \] \[ =20 \]
Step 3: Find the number of valid triangles.
\[ N={}^ {10}C_3-{}^6C_3 \] \[ N=120-20 \] \[ N=100 \]
Step 4: Final conclusion.
Therefore, \[ \boxed{100} \]
A triangle is formed by selecting \(3\) non-collinear points.
Total ways to select \(3\) points from \(10\) points: \[ {}^{10}C_3 \] \[ {}^{10}C_3=\frac{10\cdot 9\cdot 8}{3\cdot 2\cdot 1} \] \[ =120 \]
Step 2: Subtract the invalid selections.
Since \(6\) points are collinear, choosing any \(3\) points from these \(6\) points will not form a triangle.
Number of invalid selections: \[ {}^6C_3 \] \[ {}^6C_3=\frac{6\cdot 5\cdot 4}{3\cdot 2\cdot 1} \] \[ =20 \]
Step 3: Find the number of valid triangles.
\[ N={}^ {10}C_3-{}^6C_3 \] \[ N=120-20 \] \[ N=100 \]
Step 4: Final conclusion.
Therefore, \[ \boxed{100} \]
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