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 -

  • A

    $\left( {\begin{array}{*{20}{c}}
      {60} \\ 
      {50} 
    \end{array}} \right)$

  • B

    $\left( {\begin{array}{*{20}{c}}
      {60} \\ 
      {30} 
    \end{array}} \right)$

  • C

    $\left( {\begin{array}{*{20}{c}}
      {60} \\ 
      {20} 
    \end{array}} \right)$

  • D

    $\left( {\begin{array}{*{20}{c}}
      {30} \\ 
      {10} 
    \end{array}} \right)\left( {\begin{array}{*{20}{c}}
      {30} \\ 
      {20} 
    \end{array}} \right)$

Similar Questions

The sum of the coefficients of three consecutive terms in the binomial expansion of $(1+ x )^{ n +2}$, which are in the ratio $1: 3: 5$, is equal to

  • [JEE MAIN 2023]

If the Coefficient of $x^{30}$ in the expansion of $\left(1+\frac{1}{x}\right)^6\left(1+x^2\right)^7\left(1-x^3\right)^8 ; x \neq 0$ is $\alpha$, then $|\alpha|$ equals

  • [JEE MAIN 2024]

If ${S_n} = \sum\limits_{r = 0}^n {\frac{1}{{^n{C_r}}}} $ and ${t_n} = \sum\limits_{r = 0}^n {\frac{r}{{^n{C_r}}}} $, then $\frac{{{t_n}}}{{{S_n}}}$ is equal to

  • [AIEEE 2004]

If ${(1 - x + {x^2})^n} = {a_0} + {a_1}x + {a_2}{x^2} + .... + {a_{2n}}{x^{2n}}$, then ${a_0} + {a_2} + {a_4} + .... + {a_{2n}} = $

If $^{20}{C_1} + \left( {{2^2}} \right){\,^{20}}{C_3} + \left( {{3^2}} \right){\,^{20}}{C_3} + \left( {{2^2}} \right) + ..... + \left( {{{20}^2}} \right){\,^{20}}{C_{20}} = A\left( {{2^\beta }} \right)$, then the ordered pair $(A, \beta )$ is equal to

  • [JEE MAIN 2019]