A set contains $(2 n + 1)$ elements. If the number of subsets that contain at most $n$ elements is $4096$ , then the value of $n$ is:

Answer (Detailed Solution Below)View answer ▾

Answer: Option D — 6

Detailed Solution

Since $(\frac{2 n + 1}{k}) = (\frac{2 n + 1}{2 n + 1 - k})$ , subsets with at most $n$ elements are exactly half of all $2^{2 n + 1}$ subsets.

Their number is $2^{2 n} = 4096 = 2^{12}$ .

$2 n = 12$ , so $n = 6$ .

✓ Check

For

n = 6

(\frac{13}{0}) + (\frac{13}{1}) + \dots + (\frac{13}{6}) = 1 + 13 + 78 + 286 + 715 + 1287 + 1716 = 4096

⚠ Common trap

Equating the full total

2^{2 n + 1}

to 4096 gives

n = 5.5

. And (C) 15 would need

2^{30}

subsets, far more than 4096.

Quick tip

$\sum_{k = 0}^{n} (k 2 n + 1) = 2^{2 n} = 4096 \Rightarrow n = 6$

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Set Theory — Preparation

This question tests your understanding of Set Theory, a key topic in the Quantitative Aptitude section. The detailed solution above uses a step-by-step approach to show the reasoning, and where available, a shortcut that helps you solve similar questions in under 90 seconds during the actual exam.

A set is a well-defined collection of objects. Set theory covers types of sets (finite, infinite, empty, singleton, equal and equivalent sets), subsets and power sets, and operations on sets such as union, intersection, difference and complement. It also covers De Morgan's laws and the cardinality formula n(A union B) = n(A) + n(B) - n(A intersection B). In entrance exams, the most common questions are word problems about people who like one, two or three activities, solved with Venn diagrams and the cardinality formula. Questions also ask for the number of subsets of a set (2^n for a set with n elements) and test set notation and properties. Practise the set theory questions below. Each comes with a step-by-step solution and a Venn diagram where it helps.

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A set contains $(2 n + 1)$ elements. If the number of subsets that contain at most $n$ elements is $4096$ , then the value of $n$ is:

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Set Theory — Preparation

A set contains (2n+1) elements. If the number of subsets that contain at most n elements is 4096, then the value of n is:

💬 Discussions on this question

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Set Theory — Preparation

A set contains $(2 n + 1)$ elements. If the number of subsets that contain at most $n$ elements is $4096$ , then the value of $n$ is: