Permutations & Combinations
Permutation and Combination
star_batch_jee_advanced_2025
Grade None
Question:
$m$ points on one straight line are joined to $n$ points on another straight line. The number of points of intersection of the line segments thus formed (not lying on given two lines) is:
$^mC_2 \ ^nC_2$
$\frac{mn(m-1)(n-1)}{4}$
$\frac{^mC_2 \ ^nC_2}{2}$
$^mC_2 + ^nC_2$
Step-by-Step Solution
Key Concept: Two line segments connecting opposite lines intersect if and only if their endpoints are in opposite relative order, giving exactly $^mC_2 \cdot ^nC_2$ intersection points.
To find intersection points of line segments (not on the given lines), we need two line segments that intersect. Each line segment connects a point from one line to a point on the other. Two such segments intersect if and only if their endpoints are in opposite order on the two lines. Choose 2 points from the $m$ points: $^mC_2$ ways, and 2 points from the $n$ points: $^nC_2$ ways. For any such selection of 2 points on each line, exactly one pair of line segments formed will intersect inside (by the crossing property). Therefore, the number of intersection points is $^mC_2 \cdot ^nC_2 = \frac{m(m-1)}{2} \cdot \frac{n(n-1)}{2} = \frac{mn(m-1)(n-1)}{4}$.
Correct Answer: 1,2