Problem 32 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

vSuppose \(n(\geq 3)\) persons are sitting in a row. Two of them are selected at random. The probability that they are not together is (A) \(1-\frac{2}{n}\) (B) \(\frac{2}{n-1}\) (C) \(1-\frac{1}{n}\) (D) none of these

Step-by-Step Solution

Answer

The answer is (A) \(1 - \frac{2}{n}\).

Step 1: Determine Total Ways to Select Two Persons

First, calculate the total number of ways to select two people from \(n\) people. This can be done using the combination formula \( \binom{n}{2} = \frac{n(n-1)}{2} \).

Step 2: Calculate Favorable Ways When They Are Together

If two people are selected together, they can be considered as a single entity or block. So, treat these two people as one and the rest as \(n-2\) individuals, making this problem equivalent to arranging \(n-1\) entities. These two people can be selected in \(n-1\) different line-ups (with the block sliding through them).

Step 3: Calculate Probability for Persons Not Being Together

The probability that two persons are not together is calculated by subtracting the probability that they are together from 1. So, find the probability they are not together: \( 1 - \frac{n-1}{\binom{n}{2}} = 1 - \frac{2(n-1)}{n(n-1)} = 1 - \frac{2}{n} \).

Step 4: Compare to Given Options

Review the answer derived, \(1 - \frac{2}{n}\), and compare it to the listed options. Option (A) is \(1 - \frac{2}{n}\), which matches our result.

Key Concepts

Combination formula

The combination formula is an essential tool in probability and statistics. It helps us determine the number of ways to choose a subset of items from a larger set, where the order of selection doesn't matter.
In mathematical terms, the combination formula is expressed as \( \binom{n}{r} \). This notation stands for "n choose r," and it calculates the number of combinations of \( r \) items taken from \( n \) items.

For example, to choose two people from a group of \( n \), we use \( \binom{n}{2} = \frac{n(n-1)}{2} \). This means if there are 5 people, you can select a pair in 10 different ways: \( \binom{5}{2} = \frac{5 \times 4}{2} = 10 \).

The combination formula is critical when we need to explore different group formations, determine possible pairs, or make decisions that don't depend on sequence.

Probability calculation

Probability calculation is the process of quantifying the likelihood of an event occurring. It's usually expressed as a number between 0 and 1, where 0 means the event cannot happen, and 1 means it certainly will.
For example, the probability \( P(A) \) of an event \( A \) is given by:

• the number of favorable outcomes
• divided by the total number of possible outcomes

To illustrate with our problem: we first calculate the total ways two persons can be chosen using \( \binom{n}{2} \). Then, the equivalent favorable outcome, when these two are together, is determined by \( n-1 \).

We subtract the probability that two people are together from 1 to find the probability they are not together.
The expression for the latter would be:
\[ 1 - \frac{n-1}{\binom{n}{2}} = 1 - \frac{2}{n} \]
This formula helps us understand the chances of choosing two people who are not sitting together from a group.

Arrangement problem

An arrangement problem challenges us to count the number of ways to order or organize distinct individuals or items. This is key in cases where the structure matters in deriving solutions.
In our problem, if two people must sit together, they are treated like one block. Thus, the task transforms into arranging \( n-1 \) blocks (including the block of two), not just \( n \) individuals.

Think of two friends on a bench. If you agree to keep them side by side, they effectively become a single entity. However, when calculating possibilities, this bundled pair can be arranged within the sequence just as any single item can.

• Identify the block as one unit.
• Add the rest of the individuals to form \( n-1 \) units.

By considering these blocks, the original seating converts to an arrangement problem of \( n-1 \), simplifying our calculations.
This concept shines light on many structural issues, such as seating arrangements at events, organizing objects in real-life scenarios, or crafting permutations in math puzzles. Understanding arrangement problems is crucial for managing group dynamics in ordered tasks.