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Question #41705

Difficulty: 8/10 | 🟢 Free

The Question

A biased die has its faces painted with three colors: Red, Blue, and Green. The probability of rolling a Red face is $\frac{1}{2}$ , a Blue face is $\frac{1}{3}$ , and a Green face is $\frac{1}{6}$ . In a game, a player rolls this die $6$ times independently. For each Red face rolled, the player scores $1$ point; for each Blue face, $2$ points; and for each Green face, $0$ points. What is the probability that the player scores exactly $8$ points in the $6$ rolls?

Options

A.
$\frac{785}{3888}$
B.
$\frac{85}{432}$
C.
$\frac{125}{648}$
D.
$\frac{265}{1296}$

Check Your Answer

Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Specialist

Targeted practice for niche, high-difficulty concepts that frequently appear in JEE Advanced exams.

Concepts Behind This Question

Sample Space Partitioning|Bernoulli Trials Logic|Multinomial Theorem

Why This Question Is Tricky

Common Mistakes

This question tests the transition from simple binomial probability to multinomial sample space partitioning.

The primary trap lies in the shifting indices of the multinomial expansion when subject to a linear constraint ( $x+2y=8$ ).

Students commonly fail to systematically bound the variables (e.g., $z \le 2$ ), leading them to miss the edge case $(0,4,2)$ where the zero-point outcome occurs twice.

Additionally, students might incorrectly attempt to group outcomes to force a binomial model, which fails because the point values (1, 2, 0) do not map symmetrically to the probabilities.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Probability - Class 12
Concept	Sample Space Partitioning
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free