A bag contains, $7$ different Black balls .and $10$ different Red balls, if one by one ball are randomely drawn untill all black balls are not drawn, then probability that this process is completed in $12 ^{th}$ draw, is equal to 

  • A

    $\frac{{^7{C_6}{\,^{10}}{C_6}}}{{^{17}{C_{12}}}} - \frac{{^1{C_1}}}{{^5{C_1}}}$

  • B

    $\frac{{^7{C_6}{\,^{10}}{C_5}}}{{^{17}{C_{11}}}} - \frac{{^1{C_1}}}{{^6{C_1}}}$

  • C

    $\frac{{^7{C_6}{\,^{10}}{C_10}}}{{^{17}{C_{11}}}} - \frac{{^1{C_1}}}{{^6{C_1}}}$

  • D

    None

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  • [JEE MAIN 2022]