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Question #40589

Difficulty: 8/10 | 🟢 Free

The Question

Let $N$ be the number of ways to divide a set of 8 distinct points into two equal groups. Consider two skew lines $L_1$ and $L_2$ in $\mathbb{R}^3$ , where $L_1$ passes through the point $(N, 0, 0)$ and is parallel to the vector $3\hat{i} - 2\hat{j}$ , and $L_2$ passes through the point $(0, N, 0)$ and is parallel to the vector $3\hat{i} - \hat{k}$ . The shortest distance between the lines $L_1$ and $L_2$ is:

Options

A.
$5$
B.
$10$
C.
$35$
D.
$70$

Check Your Answer

Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Bridge

Practice problems that bridge the gap between foundational theory and advanced practical application.

Concepts Behind This Question

Division into Groups|Shortest distance between skew lines|Unit Vector Normalization

Why This Question Is Tricky

Common Mistakes

This question tests the intersection of combinatorics and 3D coordinate geometry.

The primary trap lies in the combinatorial phase: students frequently forget to divide by $2!$ when partitioning distinct items into equal, unlabeled groups, leading to an incorrect parameter $N=70$ instead of $N=35$ .

This error propagates into the geometry phase, yielding an incorrect distance of 10 (Option B).

A secondary trap is the failure to normalize the cross-product vector $\vec{v}_1 \times \vec{v}_2$ when calculating the shortest distance between skew lines, which would result in an answer of 35 (Option C), or 70 (Option D) if both mistakes are made.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Permutations and Combinations - Class 11
Concept	Division into Groups
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free