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NDA 2021
Exam
NDA
Year
2021
Subject
Mathematics
Chapter
Binomial Theorem

NDA 2021 · Mathematics · Binomial Theorem

If C_0, C_1, C_2, , C_n are the coefficients in the expansion of...

If C0,C1,C2,…,CnC_0, C_1, C_2, \ldots, C_n are the coefficients in the expansion of (1+x)n(1+x)^n, then what is the value of C1+C2+C3+…+CnC_1+C_2+C_3+\ldots+C_n ?

Options

  1. A2n2^n
  2. B2n−12^n-1
  3. C2n−12^{n-1}
  4. D2n−22^n-2

Correct answer

B. 2n−12^n-1

Step-by-step solution

(1+x)n=C0+C1x+C2x2+C3x3+….+Cnxn(1+x)^n=C_0+C_1 x+C_2 x^2+C_3 x^3+\ldots.+C_n x^n For, x=1,x=1,(1+1)n=C0+C1+C2+C3+….+Cn⇒2n=1+C1+C2+C3+….+Cn\begin{aligned} & (1+1)^n=C_0+C_1+C_2+C_3+\ldots.+C_n \\ & \Rightarrow 2^n=1+C_1+C_2+C_3+\ldots.+C_n\end{aligned}{∵nC0=nCn=1}∴C1+C2+C3+…..+Cn=2n−1\left\{\because{ }^n C_0={ }^n C_n=1\right\}\therefore C_1+C_2+C_3+\ldots..+C_n=2^n-1
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