Question

There are \[10\] points in the plane, no three are collinear, except 4 which are collinear. All points are joined. Let \[L\] be the number of different straight lines and \[T\] be the number of different triangles. Then, which of the following statement is true?
A. \[T = 120\]
B. \[L = 40\]
C. \[T = 3L - 5\]
D. None of these

Answer 1

Hint: Here, the positions of \[10\] points in the plane is given. First, apply the formula of combination and find the number of lines formed by the \[10\] points when no three points are collinear. Then, find the number of lines obtained from the 4 collinear points and calculate the total number of straight lines. After that, find the triangles formed by the straight lines when no three points are collinear. Then, find the number of triangles obtained from the 4 collinear points and calculate the total number of triangles. In the end, compare the values with the given options are get the required answer.

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

Complete step by step solution: Given:
\[10\] points are present in the plane and no 3 points are collinear, except 4 which are collinear.
\[L\] be the number of different straight lines.
\[T\] be the number of different triangles.

Let’s calculate the number of straight lines passing through the given points.
We know that a straight curve passing through minimum 2 points is a line.
Thus, the number of lines passing through \[10\] points when no 3 points are collinear are:
\[{}^{10}{C_2} = \dfrac{{10!}}{{2!\left( {10 - 2} \right)!}}\]
\[ \Rightarrow {}^{10}{C_2} = \dfrac{{10!}}{{2!8!}}\]
\[ \Rightarrow {}^{10}{C_2} = \dfrac{{90}}{2}\]
\[ \Rightarrow {}^{10}{C_2} = 45\]

The number of lines formed by 4 points are:
\[{}^4{C_2} = \dfrac{{4!}}{{2!\left( {4 - 2} \right)!}}\]
\[ \Rightarrow {}^4{C_2} = \dfrac{{4!}}{{2!2!}}\]
\[ \Rightarrow {}^4{C_2} = 6\]

The number of lines formed because of 4 collinear points:
\[6 - 1 = 5\]
Therefore, the total number of lines formed are:
\[L = 45 - 5\]
\[ \Rightarrow L = 40\]

Now calculate the total number of triangles.
We know that triangle can be obtained by joining 3 points, when no point is collinear.
Thus, the number of triangles obtained from \[10\] points when no 3 points are collinear are:
\[{}^{10}{C_3} = \dfrac{{10!}}{{3!\left( {10 - 3} \right)!}}\]
\[ \Rightarrow {}^{10}{C_3} = \dfrac{{10!}}{{3!7!}}\]
\[ \Rightarrow {}^{10}{C_3} = \dfrac{{720}}{6}\]
\[ \Rightarrow {}^{10}{C_3} = 120\]

The number of triangles formed by 4 points are:
\[{}^4{C_3} = \dfrac{{4!}}{{3!\left( {4 - 3} \right)!}}\]
\[ \Rightarrow {}^4{C_3} = \dfrac{{4!}}{{3!1!}}\]
\[ \Rightarrow {}^4{C_3} = 4\]

Therefore, the total number of triangles formed are:
\[T = 120 - 4\]
\[ \Rightarrow T = 116\]

Option ‘B’ is correct

Note: Collinear points: The points are said to be collinear if the points lie on the same line.
Students often get confused between permutation and combination formulas.
Combination formula: \[{}^n{C_r} = \dfrac{{n!}}{{r!\left( {n - r} \right)!}}\]
Permutation formula: \[{}^n{P_r} = \dfrac{{n!}}{{\left( {n - r} \right)!}}\]