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A hotel has four elevators. One A hotel has four elevators. One of them isa freight elevator. When pressing thebutton, one of the elevators randomlyservices your floor. If you use theelevators seven times, what is theprobability that you use the freightelevator exactly four times?

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SOLUTION

The probability that you use the freight elevator exactly four times is,

[tex]P=^7C_4(\frac{1}{4})^4(\frac{3}{4})^3[/tex]

Simplify

[tex]\begin{gathered} P=\frac{7\times6\times5\times4\times3!}{(7-4)!4!}(\frac{1}{4})^4(\frac{3}{4})^3 \\ P=\frac{7\times6\times5\times4\times3!}{3!4!}(\frac{1}{4})^4(\frac{3}{4})^3=35(\frac{1}{4})^4(\frac{3}{4})^3=\frac{945}{16384} \end{gathered}[/tex]

Hence, the answer is

[tex]\frac{945}{16384}[/tex]