Question

The number of triangles that can be formed by choosing the vertices from a set of 12 points, seven of which lie on the same straight lines, is?
A.185
B.175
C.115
D.105

Answer 1

Hint: First obtain that in \[{}^{12}{C_3}\] ways we can take 3 points from 12 points, then obtain that in \[{}^7{C_3}\] way we can take 3 points from 7 points. Then subtract \[{}^7{C_3}\] from \[{}^{12}{C_3}\] to obtain the required result.

Formula Used: \[{}^n{C_r} = \dfrac{{n!}}{{r!\left( {n - r} \right)!}}\] .

Complete step by step solution: Total number of points are 12.
So, we can choose 3 points from 12 in \[{}^{12}{C_3}\] ways.
Now,
\[{}^{12}{C_3} = \dfrac{{12!}}{{3!\left( {12 - 3} \right)!}}\]
\[ = \dfrac{{12!}}{{3!.9!}}\]
=\[\dfrac{{12.11.10.9!}}{{6.9!}}\]
\[ = 20 \times 11\]
=220
Total number of collinear points are 7.
So, we can choose 3 points from 7 in \[{}^7{C_3}\] ways.
Now,
\[{}^7{C_3} = \dfrac{{7!}}{{3!\left( {7 - 3} \right)!}}\]
\[ = \dfrac{{7!}}{{3!.4!}}\]
=\[\dfrac{{7.6.5.4!}}{{6.4!}}\]
=35

We cannot form a triangle by collinear points so, subtract 35 from 220 to obtain the required result.
\[220 - 35 = 185\]

Option ‘A’ is correct

Additional Information: Combination: When we select r object from n objects, then we use a combination formula to find the number of ways to perform this and the order of selection does not matter.

Note: Sometime students do not subtract 35 from 220 to obtain the required result. But we need to subtract 35 from 220 as we can not form a triangle by collinear points.