Question

There are $12$ points $(A_1,A_2,...,A_{12})$ in a plane, where $(A_1,A_2,A_3,A_4)$ are collinear to each other and $(A_5,A_6,A_7,A_8)$ are collinear to each other. If no points other than these two set of points are collinear, then the total number of straight lines that can be formed using these $12$ points is

• A. $54$
• B. $56$
• C. $60$
• D. $62$

Answer 1

The correct option is B $56$

Total number of straight lines that can be formed using these points without any restrictions is ${}^{12}{C}_2$
But four points are collinear due to which number of lines that can be formed using this points, ${}^4{C}_2$ should be subtracted but one line passing through all four points must be counted.

And same thing will be followed for the other set of collinear points.
So, the total number of straight lines is
${}^{12}{C}_2-{}^4{C}_2+1-{}^4{C}_2+1=56$