Twenty persons arrive in a town having $3$ hotels $x, y$ and $z$. If each person randomly chooses one of these hotels, then what is the probability that atleast $2$ of them goes in hotel $x$, atleast $1$ in hotel $y$ and atleast $1$ in hotel $z$ ? (each hotel has capacity for more than $20$ guests)

  • A

    $\frac{{^{18}{C_2}}}{{^{22}{C_2}}}$

  • B

    $\frac{{^{20}{C_2}{.^{18}}{C_1}{.^{17}}{C_1}{{.3}^{16}}}}{{{3^{20}}}}$

  • C

    $\frac{{^{20}{C_2}}}{{{3^2}}}$

  • D

    $\frac{{{3^{20\,}} - \,{{13.2}^{20}}\, + \,\,43}}{{{3^{20}}}}$

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