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Question #41837

Difficulty: 8/10 | 🟢 Free

The Question

For a positive integer $n$ , an integer $r$ is chosen at random from the set $\{0, 1, 2, \dots, n\}$ with equal probability. Let $P(n)$ be the probability that $\sum_{k=0}^{r} \binom{n}{k} \le 2^{n-1}$ If $n$ satisfies the equation $P(n) = \frac{n^2 - 100n + 2400}{(n+1)(n-60)}$ find the sum of all possible valid values of $n$ .

Options

A.
161
B.
221
C.
141
D.
80

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

Equiprobable Outcomes|Properties of Binomial Coefficients|Denominator Restriction

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to evaluate probabilities involving binomial sums and their vigilance regarding domain restrictions.

By setting up a probability equation that simplifies to a quadratic, students will find $n=60$ as a root for both the odd and even cases.

The trap is to accept $n=60$ (especially in the even case) without checking the original equation's domain.

Since $n-60$ is in the denominator, $n=60$ makes the probability undefined (division by zero), meaning it must be rejected.

Failing to do so leads to the incorrect sum of 221.

Quick Facts About This Question

Question Information

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