Let $\left(2 x ^{2}+3 x +4\right)^{10}=\sum \limits_{ r =0}^{20} a _{ r } x ^{ r } \cdot$ Then $\frac{ a _{7}}{ a _{13}}$ is equal to

  • [JEE MAIN 2020]
  • A

    $4$

  • B

    $32$

  • C

    $16$

  • D

    $8$

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The value of $\frac{1}{1 ! 50 !}+\frac{1}{3 ! 48 !}+\frac{1}{5 ! 46 !}+\ldots .+\frac{1}{49 ! 2 !}+\frac{1}{51 ! 1 !}$ is $.............$.

  • [JEE MAIN 2023]

The sum of coefficients in the expansion of ${(1 + x + {x^2})^n}$ is

If ${C_r}$ stands for $^n{C_r}$, the sum of the given series $\frac{{2(n/2)!(n/2)!}}{{n!}}[C_0^2 - 2C_1^2 + 3C_2^2 - ..... + {( - 1)^n}(n + 1)C_n^2]$, Where $n$ is an even positive integer, is

  • [IIT 1986]

Coefficient of $x^{64}$ in the expansion of $(x - 1)^2(x - 2)^3(x - 3)^4(x - 4)^5 .... (x - 10)^{11}$ 

If $r,k,p \in W,$ then $\sum\limits_{r + k + p = 10} {{}^{30}{C_r} \cdot {}^{20}{C_k} \cdot {}^{10}{C_p}} $ is equal to -