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Question #40607

Difficulty: 8/10 | 🟢 Free

The Question

Let $P_n$ be the probability that a random permutation of $n$ distinct objects is a derangement, which can be expressed as $\frac{D_n}{n!}$ . Let $p = \frac{D_4}{E}$ , where $E$ is the exponent of the prime $2$ in the denominator of $P_{16}$ . A biased coin with probability of heads $p$ is tossed $N$ times, where $N$ is the exponent of the prime $3$ in the denominator of $P_{27}$ . If the probability of obtaining an even number of heads is $\frac{1}{2} - \frac{1}{2 \cdot 5^k}$ , find the value of $k$ .

Options

A.
12
B.
13
C.
14
D.
15

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

Exponent of Prime in n!|Derangements (Inclusion-Exclusion)|Bernoulli Trials Logic

Why This Question Is Tricky

Common Mistakes

This question tests the synthesis of Legendre's formula, derangements, and the parity of successful outcomes in Bernoulli trials.

The primary trap lies in the binomial expansion for an even number of successes: $P_E = \frac{1 + (1-2p)^N}{2}$ .

Students often fail to account for the negative base when $p > 1/2$ (here $p = 3/5$ , so $1-2p = -1/5$ ), which dictates the sign of the second term based on the parity of $N$ .

Furthermore, calculating $p$ requires correctly evaluating the alternating inclusion-exclusion sum for $D_4$ and applying Legendre's formula for the prime exponents in the $n!$ denominators, bridging Combinatorics and Number Theory.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Permutations and Combinations - Class 11
Concept	Exponent of Prime in n!
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free