Q111 / 180 · COMEDK 2022 · Mathematics
There are 12 points in a plane out of which 3 points are collinear. How many straight lines can be drawn by joining any two of them?
- A60
- Correct answerB64
- C72
- D84
Solution
The total number of ways to select 2 points out of 12 points is given by $^{12}C_{2}$.
$^{12}C_{2} = \dfrac{12 \times 11}{2 \times 1} = 66$.
Since 3 points are collinear, joining any two of these 3 points results in the same line. The number of lines formed by these 3 points is $^{3}C_{2} = \dfrac{3 \times 2}{2 \times 1} = 3$.
These 3 lines are counted as distinct lines in the total $^{12}C_{2}$ calculation, but they actually represent only 1 single line. Therefore, we must subtract the 3 lines and add back the 1 line that represents the collinear set.
Total number of lines = $^{12}C_{2} - ^{3}C_{2} + 1 = 66 - 3 + 1 = 64$.
Answer: 64
$^{12}C_{2} = \dfrac{12 \times 11}{2 \times 1} = 66$.
Since 3 points are collinear, joining any two of these 3 points results in the same line. The number of lines formed by these 3 points is $^{3}C_{2} = \dfrac{3 \times 2}{2 \times 1} = 3$.
These 3 lines are counted as distinct lines in the total $^{12}C_{2}$ calculation, but they actually represent only 1 single line. Therefore, we must subtract the 3 lines and add back the 1 line that represents the collinear set.
Total number of lines = $^{12}C_{2} - ^{3}C_{2} + 1 = 66 - 3 + 1 = 64$.
Answer: 64
Hence, the correct answer is option: 'B'.