Q111 / 155 · WBJEE 2024 · Mathematics

If n is a positive integer, the value of $(2 n+1)^n C_0+(2 n-1)^n C_1+(2 n-3)^n C_2+\ldots .+1 \cdot{ }^n C_n$ is

  • Correct answer
    A
    $(n+1) 2^n$
  • B
    $3^{\text {n }}$
  • Correct answer
    C
    $f^{\prime}(2)$ where $f(x)=x^{n+1}$
  • D
    $(n+1) 2^{n+1}$

Solution

$\begin{aligned} & \text { Hint: } \sum_{r=0}^n(2 n+1-2 r)^n C_r \\ & =(2 n+1) \sum_{r=0}^n{ }^n C_r-2 \sum_{r=0}^n r \cdot{ }^n C_r \\ & =(2 n+1) \cdot 2^n-2 \cdot n \cdot 2^{n-1}=(n+1) \cdot 2^n \\ & \text { If } f(x)=x^{n+1} \Rightarrow f^{\prime}(2)=(n+1) \cdot 2^n\end{aligned}$

Hence, the correct answer is options: 'A', 'C'.

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