Question

What is the number of straight lines that can be formed by joining 20 points no three of which are in the same line except 4 of them which are in the same line?
A. 183
B. 186
C. 197
D. 185

Answer 1

Hint: First we will find the number of lines by 16 points using only two points. Then calculate the number of lines that were made by joining 1 point from the 16 points and 1 point from 4 point. Then add all lines to find the total number of lines.

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

Complete step by step solution:
The total number of points is 20 and 4 out of 20 points lie on the same line.
The number of points which do not lie on the same line is \[20 - 4 = 16\].
Now we will be calculating the number of lines by taking any two points from 20 points.
The number of lines by selecting two points is \[^{16}{C_2}\].
Apply the combination formula:
\[^{16}{C_2} = \dfrac{{16!}}{{2!\left( {16 - 2} \right)!}}\]
      \[ = \dfrac{{16 \cdot 15 \cdot 14!}}{{2 \cdot 14!}}\]
Cancel out \[14!\] from the denominator and numerator.
\[ = \dfrac{{16 \cdot 15}}{2}\]
\[ = 120\]
Now we can make a line by joining a point out of 16 points and a point out of 4 points.
The number of lines is \[^{16}{C_1}{ \times ^4}{C_1}\].
Apply combination formula
\[ = \dfrac{{16!}}{{1!15!}} \times \dfrac{{4!}}{{1!3!}}\]
\[ = \dfrac{{16 \times 15!}}{{1!15!}} \times \dfrac{{4 \times 3!}}{{1!3!}}\]
\[ = 16 \times 4\]
\[ = 64\]
There is one more that is made by joining the 4 points.
Therefore, the total number of lines is \[ = 120 + 64 + 1 = 185\]

Option ‘D’ is correct

Note: Because they are collinear, we can get any one line by selecting both points from 2 in 2. DO not forgot to add 1 with \[120 + 64\] as we have to get total number of lines, so \[ = 120 + 64 + 1 = 185\]