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Question #41615

Difficulty: 8/10 | 🟢 Free

The Question

Let $S = \{0, 1\}$ . Three vectors $\vec{a}, \vec{b}, \vec{c}$ in $\mathbb{R}^3$ are generated by choosing their components independently and uniformly at random from $S$ . What is the probability that the volume of the parallelepiped formed by $\vec{a}, \vec{b}, \vec{c}$ is non-zero?

Options

A.
$\frac{87}{256}$
B.
$\frac{21}{64}$
C.
$\frac{89}{256}$
D.
$\frac{43}{128}$

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

Scalar Triple Product Volume Logic|Sample Space Partitioning|Mutually Exclusive vs Independent|Equiprobable Outcomes

Why This Question Is Tricky

Common Mistakes

This question tests the distinction between linear independence over $\mathbb{R}$ versus finite fields (like $\mathbb{F}_2$ ).

A common trap is assuming that the span of two linearly independent vectors over $\{0,1\}^3$ always contains exactly $2^2 = 4$ elements, leading to the incorrect calculation of $7 \times 6 \times 4 = 168$ favorable outcomes (Option B).

Students fail to recognize that for specific pairs like $(1,1,0)$ and $(1,0,1)$ , their linear combinations fall outside the restricted sample space $\{0,1\}^3$ , reducing the coplanar intersection to just 3 vectors and increasing the valid choices for the third vector.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Vector Algebra - Class 12
Concept	Scalar Triple Product Volume Logic
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free