If C_ 0 , C_ 1 , C_ 2 , . C_ n denote the binomial coefficientsin the expansion of (1+x)^ n , then the value of C_ 0 +(C_ 0 +C_ 1 )+ (C_ 0 +C_ 1 +C_ 2 )+ +(C_ 0 +C_ 1 + ..+C_ n-1 )

Correct answer: A. n · 2^ n-1 aligned & C_ 0 +(C_ 0 +C_ 1 )+(C_ 0 +C_ 1 +C_ 2 )+ &+(C_ 0 +C_ 1 + . C_ n-1 ) &=n C_ 0 +(n-1) C_ 1 +(n-2) C_ 2 + C_ n-1 &=C_ 1 +2 C_ 2 +3 C_ 3 +4 C_ 4 n C_ n =n.2^ n-1 aligned

BITSAT 2013

Exam: BITSAT
Year: 2013
Subject: Mathematics
Chapter: Binomial Theorem

BITSAT 2013 · Mathematics · Binomial Theorem

If C_ 0 , C_ 1 , C_ 2 , . C_ n denote the binomial coefficientsin...

\mathrm{C}_{0}, \mathrm{C}_{1}, \mathrm{C}_{2}, \ldots \ldots \ldots \ldots. \mathrm{C}_{\mathrm{n}}

denote the binomial coefficientsin the expansion of

(1+\mathrm{x})^{\mathrm{n}}

, then the value of $\mathrm{C}_{0}+\left(\mathrm{C}_{0}+\mathrm{C}_{1}\right)+ \left(\mathrm{C}_{0}+\mathrm{C}_{1}+\mathrm{C}_{2}\right)+\ldots+\left(C_{0}+C_{1}+\ldots..+C_{n-1}\right)

Options

A $\mathrm{n} \cdot 2^{\mathrm{n}-1}$
B $\mathrm{n} \cdot 2^{\mathrm{n}}$
C $(n-1) \cdot 2^{n-1}$
D $(n-1) \cdot 2^{n}$

Correct answer

A. $\mathrm{n} \cdot 2^{\mathrm{n}-1}$

Step-by-step solution

\begin{aligned} & \mathrm{C}_{0}+\left(\mathrm{C}_{0}+\mathrm{C}_{1}\right)+\left(\mathrm{C}_{0}+\mathrm{C}_{1}+\mathrm{C}_{2}\right)+\ldots \ldots \ldots \ldots \ldots \\ &+\left(\mathrm{C}_{0}+\mathrm{C}_{1}+\ldots \ldots \ldots. \mathrm{C}_{\mathrm{n}-1}\right) \\ &=\mathrm{n} \mathrm{C}_{0}+(\mathrm{n}-1) \mathrm{C}_{1}+(\mathrm{n}-2) \mathrm{C}_{2}+\ldots \ldots \mathrm{C}_{\mathrm{n}-1} \\ &=\mathrm{C}_{1}+2 \mathrm{C}_{2}+3 \mathrm{C}_{3}+4 \mathrm{C}_{4} \ldots \ldots \mathrm{n} \mathrm{C}_{\mathrm{n}}=\mathrm{n}.2^{\mathrm{n}-1} \end{aligned}

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