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

Question

\(A\) set contains \((2 n+1)\) elements. The number of subsets of the set which contain at most \(n\) elements is (A) \(2^{n}\) (B) \(2^{n+1}\) (C) \(2^{2 n-1}\) (D) \(2^{2 n}\)

Step-by-Step Solution

Answer

(D) \(2^{2n}\)

Step 1: Calculate Total Number of Subsets

The total number of subsets of a set with \((2n+1)\) elements is given by \(2^{2n+1}\). This is because each element can either be included or not included in a subset.

Step 2: Calculate Subsets with More than n Elements

Subsets that have more than \(n\) elements range from \((n+1)\) to \((2n+1)\) elements. By symmetry, the number of subsets with at most \(n\) elements is equal to the number of subsets with more than \(n\) elements.

Step 3: Find Subsets with Exactly (n+1) to (2n+1) Elements

Since the subsets are symmetric around \((2n+1)/2 = n+0.5\), the number of subsets with more than \(n\) elements is equal to the subsets with at most \(n\) elements.

Step 4: Apply Symmetry to Get the Subset Count

Since the subsets are symmetric, the number of subsets with at most \(n\) elements is half of the total number of subsets. Thus, the number is \(\frac{2^{2n+1}}{2} = 2^{2n}\).

Key Concepts

Number of Subsets

Understanding the number of subsets of a set is crucial in combinatorics. In its simplest form, when you have a set with a certain number of elements, every element can either be in a subset or not. This gives each element two choices: include it in the subset or exclude it. Therefore, for a set with \(n\) elements, there are \(2^n\) different possible subsets.

For example, for a set with \(3\) elements \(\{a, b, c\}\), the subsets include: the empty set, \(\{a\}\), \(\{b\}\), \(\{c\}\), \(\{a, b\}\), \(\{a, c\}\), \(\{b, c\}\), and \(\{a, b, c\}\). That's exactly \(2^3 = 8\) subsets.

So, when calculating subsets, remember each element gives you two options - doubling the possibilities.

Symmetry in Sets

Symmetry plays a fascinating role in understanding sets, especially when working out subsets with a particular number of elements. The principle of symmetry in sets tells us that the subsets of a set can be split evenly around the halfway mark of the total elements.

For instance, if you have subsets of a set with size \((2n+1)\), you'll find symmetry around the midpoint, allowing simplifications of calculations.

To illustrate, in a set with \((2n+1)\) elements, subsets containing at most \(n\) elements are mirrored by subsets that have more than \(n\) elements.

This means if you know the count of subsets on one side of the midpoint, the count is the same on the other side. In the case from the problem, subsets with at most \(n\) elements are equal in number to those with more than \(n\) elements.

Elements in a Set

The elements in a set are fundamental components in set theory. Each element can be thought of as a distinct object or item within the collection that is the set.

When determining characteristics like the number of possible subsets, the count of elements is paramount. It's often denoted by \(n\) for general calculations, and this directly influences the complexity or number of total possible combinations (subsets).

In particular problems, like the one provided, specific forms such as \((2n+1)\) can help identify structured problems to apply advanced concepts like symmetry.

By knowing the number of elements, tools like combinatorial analysis and mirror symmetry can be used effectively, leading to elegant solutions and deeper understanding of the properties involved.