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NDA 2023
Exam
NDA
Year
2023
Subject
Mathematics
Chapter
Binomial Theorem

NDA 2023 · Mathematics · Binomial Theorem

A set S contains (2n +1) elements. There are 4096 subset of S which...

A set S contains (2n +1) elements. There are 4096 subset of S which contain at most n elements. what is n equal to

Options

  1. A5
  2. B6
  3. C7
  4. D8

Correct answer

B. 6

Step-by-step solution

 Since, n(s)=2n+1Now, 2n+1C0+2n+1C1+2n+1C2+….+2n+1Cn+2n+1Cn+1+2n+1Cn+2+…..+2n+1C2n+1=22n+1⇒2(2n+1C0+2n+1C1+….+2n+1Cn)=22n+1=4096⇒22n=212⇒2n=12⇒n=6\begin{aligned} & \text{ Since, } \mathrm{n}(\mathrm{s})=2 \mathrm{n}+1 \\ & \text{Now, }{ }^{2 \mathrm{n}+1} \mathrm{C}_0+{ }^{2 \mathrm{n}+1} \mathrm{C}_1+{ }^{2 \mathrm{n}+1} \mathrm{C}_2+\ldots.+{ }^{2 \mathrm{n}+1} \mathrm{C}_{\mathrm{n}}+ \\ & { }^{2 \mathrm{n}+1} \mathrm{C}_{\mathrm{n}+1}+{ }^{2 \mathrm{n}+1} \mathrm{C}_{\mathrm{n}+2}+\ldots..+{ }^{2 \mathrm{n}+1} \mathrm{C}_{2 \mathrm{n}+1}=2^{2 \mathrm{n}+1} \\ \Rightarrow & 2\left({ }^{2 \mathrm{n}+1} \mathrm{C}_0+{ }^{2 \mathrm{n}+1} \mathrm{C}_1+\ldots.+{ }^{2 \mathrm{n}+1} \mathrm{C}_{\mathrm{n}}\right)=2^{2 \mathrm{n}+1}=4096 \\ \Rightarrow & 2^{2 \mathrm{n}}=2^{12} \Rightarrow 2 \mathrm{n}=12 \Rightarrow \mathrm{n}=6\end{aligned}
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