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NDA 2022
Exam
NDA
Year
2022
Subject
Mathematics
Chapter
Binomial Theorem

NDA 2022 · Mathematics · Binomial Theorem

What is ∑_ r=0 ^n 2^r C(n, r) equal to?

What is ∑r=0n2rC(n,r)\sum_{r=0}^n 2^r C(n, r) equal to?

Options

  1. A2n2^n
  2. B3n3^{\mathrm{n}}
  3. C22n2^{2 n}
  4. D32n3^{2 n}

Correct answer

B. 3n3^{\mathrm{n}}

Step-by-step solution

∑r=0n2rnCr=nC020+nC121+nC222+…+nC22n=(1+2)n=3n\begin{aligned} & \sum_{r=0}^n 2^r{ }^{\mathrm{n}} C r={ }^{\mathrm{n}} C_0 2^0+{ }^{\mathrm{n}} C_1 2^1+{ }^{\mathrm{n}} C_2 2^2+\ldots+{ }^{\mathrm{n}} C_2 2^{\mathrm{n}} \\ & =(1+2)^{\mathrm{n}}=3^{\mathrm{n}}\end{aligned}
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