Question

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?

• A. 60
• B. 64
• C. 72
• D. 84

Hint:

If \( r \) points are collinear, they contribute only 1 line instead of \( {}^rC_2 \) lines. Subtract \( {}^rC_2 - 1 \) from the total.

Answer 1

The Correct Option is B

Step 1: Total number of ways of selecting any two points out of 12 points is given by \({}^{12}C_2\):
\[ {}^{12}C_2 = \frac{12 \times 11}{2} = 66 \]
Step 2: Since 3 points are collinear, the 3 points lie on the same line. So the 3 points will contribute only 1 line instead of \( {}^3C_2 = 3 \) lines. So, we subtract the extra lines:
\[ 66 - ({}^3C_2 - 1) = 66 - (3 - 1) = 66 - 2 = 64 \]
Thus, the number of straight lines that can be drawn is 64.