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Question #41017

Difficulty: 8/10 | 🟢 Free

The Question

Four identical balls are distributed randomly into three distinct boxes, such that each ball is independently placed into any of the boxes with equal probability. Let $P_0$ be the probability that the first box is empty, and $P_2$ be the probability that the first box contains exactly two balls. Find the sum of all integers $x \in [1, 10]$ that satisfy the inequality: $\frac{x^2 - 11x + 28}{P_0 - P_2} \ge 0$

Options

A.
22
B.
44
C.
11
D.
55

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

Equiprobability Fallacy|Distributing Identical Items (Stars and Bars)|Cross Multiplication Trap

Why This Question Is Tricky

Common Mistakes

This question targets the Equiprobability Fallacy and the Cross Multiplication Trap.

Students often incorrectly use 'Stars and Bars' to calculate probabilities for identical items, assuming the $\binom{n+r-1}{r-1}$ macrostates are equally likely.

This fallacy yields $P_0 = 5/15$ and $P_2 = 3/15$ , making the denominator incorrectly positive ( $2/15$ ).

Alternatively, even with correct probabilities, students may fall for the Cross Multiplication Trap by multiplying the inequality by the negative denominator ( $-8/81$ ) without flipping the inequality sign.

Both conceptual errors lead to solving $x^2 - 11x + 28 \ge 0$ , resulting in the trap answer 44.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Probability - Class 11
Concept	Equiprobability Fallacy
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free