Problem 16 - Algebra 2 Course in Mathematics for the IIT-JEE and Other Engineering Exams

Question

If a party of \(n\) persons sits at a round table then the odds against two specified individuals sitting next to each other are (a) \(2:(n-3)\) (b) \((n-3): 2\) (c) \((n-2): 2\) (d) \(2:(n-2)\)

Step-by-Step Solution

Answer

Option (b): \\(n-3): 2\\.

Step 1: Understand the Arrangements

First, note that in a round table configuration, circular permutations are used. For any set of \( n \) people, fixing one as a reference due to the circular arrangement, there are \((n-1)!\) ways to arrange the remaining \(n-1\) people.

Step 2: Count Target Arrangements

Consider the case where two specified individuals, say A and B, need to sit next to each other. Treat them as a single unit or 'block'. This 'block' joins the other \((n-2)\) individuals, forming \((n-1)\) units to arrange around the table. The total arrangements for these units is \((n-2)!\). However, since A and B within the block can switch with each other, multiply by \(2\). So you have \(2\times (n-2)!\).

Step 3: Determine Total Arrangements

As stated initially, the total number of arrangements for \(n\) persons around a round table is \((n-1)!\).

Step 4: Find Favorable Outcomes

Favorable outcomes for A and B sitting next to each other are calculated in Step 2, i.e., \(2 \times (n-2)!\).

Step 5: Calculate Odds Against

The odds against the event is calculated as the ratio of unfavorable outcomes to favorable outcomes. Unfavorable outcomes where A and B are not sitting next to each other is given by \((n-1)! - 2 \times (n-2)!\). Hence the odds against A and B sitting next to each other is, \(\frac{(n-1)! - 2 \times (n-2)!}{2 \times (n-2)!}\). Simplifying gives \(\frac{2 \times (n-3)!}{2}\), or \((n-3):2\).

Step 6: Choose the Correct Option

The odds calculated are \((n-3):2\), which matches option (b).

Key Concepts

Circular Permutations

Circular permutations differ from linear permutations. In a circular permutation, the arrangement loops around, making some arrangements indistinguishable from one another.
This specific circular aspect is evident in scenarios like round tables. The key point about circular permutations is that one position is usually fixed for comparison, leaving the rest of the items to be permuted.

• For instance, if you have 5 people and you arrange them in a circle, fixing one person means you only need to arrange the remaining 4 people. This results in \( (n-1)! \) permutations, where 'n' is the total number of people.
• This differs from linear permutations which would be \(n!\) because every position is distinct.

When considering circular permutations for two specific individuals to be together, treat them as a single unit or block, then arrange the block with the others.

Combinatorics

Combinatorics is the branch of mathematics concerning the counting, arrangement, and combination of objects.
It provides a foundation for understanding problems involving arrangements and groupings.One part of combinatorial problems involves arrangements where you determine how many ways you can order a set of objects.

• The most straightforward example is arranging books on a shelf: if you have 3 books, the number of arrangements is \(3! = 6\) because that's 3 times 2 times 1.
• In a circular setup, combinatorics helps to find different ways objects can be arranged considering the rotational symmetry.

For circular permutation problems, combinatorial logic helps simplify calculations by fixing one position and arranging the rest, thus reducing a circular problem to a simpler linear problem.

Probability

Probability is the measure of how likely an event is to occur. In the context of arranging people at a table, probability helps to determine the likelihood that a specific arrangement occurs.
For solving arrangement problems with specific conditions, like certain people sitting together or apart, probability plays a crucial role.

• To calculate the probability of an event using arrangements, you divide the number of favorable outcomes by the total number of possible outcomes.
• In calculating odds against an event, as in the problem, find the number of favorable arrangements (specific people sitting together) and compare this with all possible arrangements, subtracting the favorable ones for the odds against.

Using probability to address specific constraints in arrangements allows for a clear understanding of how likely certain seating orders are within group settings.

Arrangements

Arrangements refer to the various ways in which a set of objects or people can be organized. In probability and combinatorics, arrangements can either be linear or circular.
While linear arrangements are straightforward, circular ones add a layer of complexity often addressed through fixing a reference point. When evaluating arrangements like those at a circular table, you can use:

• "Block Understanding" - Treating specific individuals as a single unit when they need to be grouped.
• "Fix and Permute" - Fixing one person or object as a starting point, then permuting the rest.

Understanding arrangements allows people to approach seemingly complex problems methodically by breaking them into manageable steps.
By first analyzing how many total arrangements exist, it’s possible to then filter those that match certain criteria, such as two people sitting together or apart, thereby honing in on the specific arrangement needed.