The sum of coefficients in the expansion of ${(x + 2y + 3z)^8}$ is

  • A

    ${3^8}$

  • B

    ${5^8}$

  • C

    ${6^8}$

  • D

    None of these

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The value of $\frac{{{C_1}}}{2} + \frac{{{C_3}}}{4} + \frac{{{C_5}}}{6} + .....$ is equal to

Let $C _{ r }$ denote the binomial coefficient of $x ^{ r }$ in the expansion of $(1+x)^{10}$. If $\alpha, \beta \in R$. $C _{1}+3 \cdot 2 C _{2}+5 \cdot 3 C _{3}+\ldots$ upto $10$ terms $=\frac{\alpha \times 2^{11}}{2^{\beta}-1}\left( C _{0}+\frac{ C _{1}}{2}+\frac{ C _{2}}{3}+\ldots . .\right.$ upto 10 terms $)$ then the value of $\alpha+\beta$ is equal to

  • [JEE MAIN 2022]

The sum of last eight consecutive coefficients in the expansion of $(1+x)^{15}$ is

$^n{C_0} - \frac{1}{2}{\,^n}{C_1} + \frac{1}{3}{\,^n}{C_2} - ...... + {( - 1)^n}\frac{{^n{C_n}}}{{n + 1}} = $