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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)
$\frac{{^{18}{C_2}}}{{^{22}{C_2}}}$
$\frac{{^{20}{C_2}{.^{18}}{C_1}{.^{17}}{C_1}{{.3}^{16}}}}{{{3^{20}}}}$
$\frac{{^{20}{C_2}}}{{{3^2}}}$
$\frac{{{3^{20\,}} - \,{{13.2}^{20}}\, + \,\,43}}{{{3^{20}}}}$
Solution
$A \rightarrow x \leq 1$
$\mathrm{B} \rightarrow \mathrm{y}=0$
$\mathrm{C} \rightarrow \mathrm{z}=0$
$n(A \cup B \cup C)$
$=20\left(2^{19}\right)+1\left(2^{20}\right)+1\left(2^{20}\right)+1\left(2^{20}\right)$
$-((21 \times 1))-1-(21 \times 1)$
Total cases $=3^{20}$
Ans. $=\frac{3^{20}-13 \cdot 2^{20}+43}{3^{20}}$
