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Question #41832

Difficulty: 8/10 | 🟢 Free

The Question

Let $S$ be the sample space of all integers $k \in [-50, 50]$ such that $\\binom{100}{50+k}$ is odd. A number $k$ is chosen uniformly at random from $S$ . Let $E$ be the expected value of the random variable $X = k^3 \\sin\\left(\\frac{\\pi k}{4}\\right) \\ln\\left(\\frac{100-k}{100+k}\\right) + \\frac{12}{1+3^k}$ Find the value of $E \\times |S|$ .

Options

A.
48
B.
60
C.
24
D.
0

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

Combinatorial Probability|Exponent of Prime in n!|Even and Odd Function Symmetry

Why This Question Is Tricky

Common Mistakes

The question integrates combinatorial probability, number theory (Lucas' Theorem/Legendre's Formula), and algebraic symmetry.

The core trap lies in the evaluation of the expected value over the sample space S.

Students often recognize the odd function f(k) and the symmetric interval, incorrectly assuming the entire expectation is 0 (Option D).

Furthermore, to find the total sum E × |S|, students must correctly apply Lucas' Theorem to determine the cardinality of S.

Failing to recognize the base-2 subset construction will lead to an incorrect |S|, trapping them in wrong proportional answers.

Quick Facts About This Question

Question Information

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