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NDA 2013
Exam
NDA
Year
2013
Subject
Mathematics
Chapter
Binomial Theorem

NDA 2013 · Mathematics · Binomial Theorem

What is ∑_ r=0 ^n C(n, r) equal to?

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

Options

  1. A2n−12^n-1
  2. Bn
  3. Cn!
  4. D2n2^n

Correct answer

D. 2n2^n

Step-by-step solution

(1+x)n=nC0+nC1x+nC2x2+……+nCnxnfor x=1(2)n=nC0+nC1+nC2+nC3+……+nCn∑r=0nC(n,r)=(2)n.\begin{aligned} &(1+x)^n={ }^n \mathrm{C}_0+{ }^n \mathrm{C}_1 x+{ }^n \mathrm{C}_2 x^2+\ldots \ldots+{ }^n \mathrm{C}_n x^n \\ & \text{for } \quad x=1 \\ &(2)^n={ }^n \mathrm{C}_0+{ }^n \mathrm{C}_1+{ }^n \mathrm{C}_2+{ }^n \mathrm{C}_3+\ldots \ldots+{ }^n \mathrm{C}_n \\ & \sum_{r=0}^n \mathrm{C}(n, r)=(2)^n.\end{aligned}
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