Problem 55 - A Complete Resource Book in Mathematics for JEE Main

Question

Suppose \(n\) people are asked a question successively in a random order and exactly 3 of the \(n\) people know that answer. If \(n>6\), the probability that the first four of those asked do not know the answer is (A) \(\frac{{ }^{n-4} C_{4}}{{ }^{n} C_{4}}\) (B) \(\frac{{ }^{n-3} C_{4}}{{ }^{n} C_{4}}\) (C) \(\frac{1}{{ }^{n} C_{4}}\) (D) none of these

Step-by-Step Solution

Answer

The answer is (B): \(\frac{{ }^{n-3} C_{4}}{{ }^{n} C_{4}}\).

Step 1: Understand the Problem

We have a total of \(n\) people, and only 3 of them know the answer. We want to find the probability that when four people are picked at random, none of them know the answer.

Step 2: Determine the Non-Knowers

Since only 3 people out of \(n\) know the answer, the remaining \(n - 3\) people do not know the answer. We need to select 4 people from these \(n-3\) people to ensure none know the answer.

Step 3: Apply Combinatorials for Non-Knowers

Calculate the number of ways to choose 4 people from \(n-3\) non-knowers, which is \(\binom{n-3}{4}\).

Step 4: Compute Total Combinations

Calculate the total number of ways to choose any 4 people from \(n\) people, which is \(\binom{n}{4}\).

Step 5: Evaluate the Probability

The probability that the first four people selected do not know the answer is given by the ratio of ways to choose 4 non-knowers to ways to choose any 4 people: \( \frac{\binom{n-3}{4}}{\binom{n}{4}} \).

Step 6: Choose the Correct Answer

Comparing with the given options, the correct expression matches option (B).

Key Concepts

Combinatorics

Combinatorics is a fascinating area of mathematics that focuses on counting, arranging, and structuring sets or groups of elements. It allows us to calculate how many possible combinations or arrangements can be made in a variety of situations. In the context of the problem mentioned, we use combinatorial techniques to determine how many different ways we can select a group of people.
For instance, when dealing with a group of people, we might want to find out all possible ways to choose a certain number of individuals from that group. This involves using specific combinatorial methods, which help us break down the problem into smaller, more manageable parts.

• Combinatorics includes permutation (ordering items) and combination (selecting items) problems.
• It's widely applicable in fields like computer science, cryptography, and statistics.

By understanding the combinatorial foundation of the problem, we can more easily compute things like probabilities and likelihoods, which are crucial for solving mathematical problems.

Binomial Coefficient

At the heart of combinatorial problem-solving lies the binomial coefficient, often denoted as \(\binom{n}{k}\). This notation represents the number of ways to choose \(k\) elements from a set of \(n\) elements without regard to the order of selection. In simpler terms, it provides a count of possible combinations.
The binomial coefficient is calculated using the formula:\[ \binom{n}{k} = \frac{n!}{k!(n-k)!} \]where \(!\) denotes factorial, meaning the product of all positive integers up to that number.
In the problem provided, the binomial coefficient \(\binom{n-3}{4}\) describes the number of ways we can select 4 people from \(n-3\) non-knowers. This concept is crucial because it allows us to systematically determine probabilities by considering the number of favorable outcomes relative to the total number of outcomes.

Mathematical Problem Solving

Mathematical problem-solving involves a set of skills and techniques that empower us to tackle various analytical challenges. For probability problems, this includes understanding the context, identifying known quantities, and logically structuring a path to the solution.
In our given exercise, we start by understanding who knows the answer and who doesn't. This insight allows us to categorize and calculate in a structured manner. We then use logic to set up an equation that relates our known elements to what's being asked—in this case, the probability of selecting four non-knowers.

• Identifying the knowns and unknowns helps in structuring the problem.
• Combining the right principles (like the binomial coefficient) aids in simplifying the solution.

By drawing on combinatorial knowledge and structured logical reasoning, we can break down complex problems into understandable parts. This makes solving mathematical exercises not only more accessible but also much more enjoyable.