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Question #41821

Difficulty: 8/10 | 🟢 Free

The Question

Let $S = \{(x, y, z) \mid x, y, z \in \{0, 1\}\}$ . Three vectors $\vec{u}, \vec{v}, \vec{w}$ are chosen independently and with replacement from $S$ . If $P$ is the probability that the vectors $\vec{u}, \vec{v}, \vec{w}$ are coplanar, then $P$ equals

Options

A.
$\frac{169}{256}$
B.
$\frac{43}{64}$
C.
$\frac{81}{256}$
D.
$\frac{21}{64}$

Check Your Answer

Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Monster

Ultra-high difficulty questions designed to test your limit, stamina, and deep conceptual understanding.

Concepts Behind This Question

Sampling With/Without Replacement|Determinant as Geometric Volume|Collinearity vs Coplanarity|Condition of Collinearity

Why This Question Is Tricky

Common Mistakes

This question tests the intersection of probability, linear algebra, and combinatorics.

The primary trap lies in equating linear dependence over the real numbers $\mathbb{R}$ with linear dependence over the finite field $\mathbb{F}_2$ .

Students often calculate the number of non-invertible matrices over $\mathbb{F}_2$ ( $512 - 168 = 344$ ) and incorrectly assume all these have a determinant of zero over $\mathbb{R}$ , leading to the trap answer 43/64.

They fail to account for the 6 matrices with a determinant of $\pm 2$ , which are singular over $\mathbb{F}_2$ but non-singular over $\mathbb{R}$ .

A secondary trap is attempting to manually count coplanar distinct vectors while forgetting the 'with replacement' condition that allows identical rows.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Probability - Class 12
Concept	Sampling With/Without Replacement
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free