Question

If $C_{0}, C_{1}, C_{2}, \ldots \ldots \ldots \ldots C_{n}$ denote the binomial coefficients in the expansion of $(1+x)^{n}$, then the value of $C_{0}+\left(C_{0}+C_{1}\right)+\left(C_{0}+C_{1}+C_{2}\right)+\ldots .$ $+\left(C_{0}+C_{1}+\ldots .+C_{n-1}\right)$

• A. $n.2^{n-1}$
• B. $n.2^{n}$
• C. $(n -1) .2^{n -1 }$
• D. $(n - 1) .2^{n}$

Answer 1

Answer: (A)

$C_{0}+\left(C_{0}+C_{1}\right)+\left(C_{0}+C_{1}+C_{2}\right)+\ldots \ldots \ldots \ldots$
$+\left(C_{0}+C_{1}+\ldots \ldots \ldots \ldots C_{n}-1\right)$
$={ }^{n} C_{0}+(n-1) C_{1}+(n-2) C_{2}+\ldots \ldots . C_{n-1}$
$=C_{1}+2 C_{2}+3 C_{3}+4 C_{4} \ldots \ldots n C_{n}=n .2^{n-1}$