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]
  • A

    $(420, 18)$

  • B

    $(380, 18)$

  • C

    $(420, 19)$

  • D

    $(380, 19)$

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  • [JEE MAIN 2021]