Problem 88 - Algebra for JEE ADVANCED

Question

The total number of ways in which \(2 n\) persons can be divided i nto \(n\) couples is a. \(\frac{2 n !}{n ! n !}\) b. \(\frac{2 n !}{(2 !)^{n}}\) c. \(\frac{2 n !}{n !(2 !)^{n}}\) d. none of these

Step-by-Step Solution

Answer

The answer is option c: \(\frac{2n!}{n!(2!)^n}\).

Step 1: Understanding the Problem

We need to find the total number of ways in which \(2n\) persons can be divided into \(n\) couples. Each couple consists of 2 persons, and the order within the couple does not matter.

Step 2: Total Permutations of 2n Persons

First, we consider the total number of ways to arrange \(2n\) persons in a line, which is given by the total number of permutations \(2n!\).

Step 3: Divide into Couples

Since the order within each couple does not matter, we divide by \(2!\) for each couple. For \(n\) couples, this step contributes \((2!)^n\) to the denominator in our calculation.

Step 4: Arrange Couples

Once the couples are formed, the order of the \(n\) couples does not matter. Therefore, we also need to divide by \(n!\) to account for the different arrangements of \(n\) indistinguishable couples.

Step 5: Calculate the Total

Putting it all together, the number of ways to form \(n\) couples from \(2n\) people is \(\frac{2n!}{n!(2!)^n}\). This matches choice \(c\) in the list of options.

Key Concepts

Permutations

Permutations are all about arranging objects where the order matters. Imagine you're organizing books on a shelf: the order in which you place them makes a difference. In mathematics, a permutation of a set is an arrangement of its members into a sequence or linear order.

• The formula for finding the number of permutations of a set of objects is represented by the factorial of the number of objects \( n! \).
• For example, if you have 3 distinct objects (let's say A, B, and C), the number of permutations is \(3! = 6\).
• The different arrangements (or permutations) would be: ABC, ACB, BAC, BCA, CAB, CBA.

When dealing with larger groups, such as the exercise involving \(2n\) people, the concept of permutations becomes especially important for understanding how many ways you can arrange these individuals in a linear sequence before sorting them into pairs or groups. Knowing permutations help us to lay out all possibilities from which we can further refine based on additional conditions like belonging to couples.

Combinations

Combinations are about selecting items where the order doesn’t matter. Unlike permutations, you focus solely on choosing objects without considering different arrangements.

• The basic formula for combinations is given by \(\binom{n}{r} = \frac{n!}{r!(n-r)!}\), where \(n\) is the total number of items to choose from, and \(r\) is the number of items to select.
• For example, if you're picking 2 out of 3 {A, B, C}, then \(\binom{3}{2} = 3\). Your choices are: AB, AC, and BC.
• This relates to the problem of finding the number of ways to choose pairs from a group of people, as in dividing \(2n\) persons into \(n\) couples. Here, once individuals are paired (order within the pair is crucial initially), within the pair, the order doesn't matter.

Combinations are essential in scenarios like our original exercise because they help calculate the number of subsets of a particular size from the available set, untangling the configurations where the order is irrelevant.

Factorials

Factorials are the foundation of both permutations and combinations. The factorial of a number \( n \), represented as \( n! \), is the product of all positive integers up to that number. Factorials grow extremely quickly with increasing \( n \).

• For example, \(5! = 5 \times 4 \times 3 \times 2 \times 1 = 120\).
• Factorials are used to calculate permutations because they account for every possible order of arrangement.
• In combinations, factorials help account for the ordering within the chosen subset itself (i.e., why we divide by additional factorials in the combination formula).

In the context of our exercise, \(2n!\) is used to consider all possible arrangements of people before accounting for the ignored order within couples. This simplicity allows us to unravel complex problems involving arrangements and selections by breaking them down into systematically manageable parts.