If $\frac{1}{n+1}{ }^n C_n+\frac{1}{n}{ }^n C_{n-1}+\ldots+\frac{1}{2}{ }^{ n } C _1+{ }^{ n } C _0=\frac{1023}{10}$ then $n$ is equal to

  • [JEE MAIN 2023]
  • A

    $6$

  • B

    $9$

  • C

    $8$

  • D

    $7$

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The coefficient of $x^{4}$ in the expansion of $\left(1+x+x^{2}+x^{3}\right)^{6}$ in powers of $x,$ is

  • [JEE MAIN 2020]

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  4 \hfill \\
  n \hfill \\ 
\end{gathered}  \right)} {\left( { - 1} \right)^n}$   is

In the polynomial $(x - 1)(x - 2)(x - 3).............(x - 100),$ the coefficient of ${x^{99}}$ is

If $n$ be a positive integer such that $n \ge 3$, then the value of the sum to $n$ terms of the series $1 . n - \frac{{\left( {n\, - \,1} \right)}}{{1\,\,!}} (n - 1) + \frac{{\left( {n\, - \,1} \right)\,\,\left( {n\, - \,2} \right)}}{{2\,\,!}} (n - 2) $$-  \frac{{\left( {n\, - \,1} \right)\,\,\left( {n\, - \,2} \right)\,\,\left( {n\, - \,3} \right)}}{{3\,\,!}} (n - 3) + ......$ is