Twelve students will be chosen at random from the 900 students at Rolling Meadows High School to serve as the Judicial Board for minor student infractions . (a) How many different Judicial Boards are possible? (b) Mariko hopes to be on the board. How many possible boards include her? (c) What is the probability (in percent) that Mariko is chosen for the board?

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$\textbf {a.}$

Selecting 12 students out of 900 regardless the order:

\begin{align*} _{900}C_{12}&=\dfrac {900!}{(900-12)! \times 12!} \\ &=\dfrac {900!}{(888)! \times 12!} \\ &\boxed {\color{#c34632} =5.4 \times 10^{26}} \\ \end{align*}

$\textbf {b.}$

If Mariko in in the board, the number of possible boards include her:

\begin{align*} _{899}C_{11}&=\dfrac {899!}{888! \times 11!} \\ &\boxed {\color{#c34632} =7.3 \times 10^{23}} \end{align*}

The probability of choosing Mariko:

\begin{align*} P(\text {Mariko})&= \dfrac {\text {Number of ways that Mariko is chosen}}{\text {Total number of ways}} \\ &=\dfrac {_{899}C_{11}}{_{900}C_{12}} \\ &=\dfrac {899!}{888! \times 11!} \div \dfrac {900!}{(888)! \times 12!} \\ &=\dfrac {899!}{888! \times 11!} \times \dfrac {(888)! \times 12!}{900!} \\ &=\dfrac {\cancel {899!}}{\cancel {888!} \times \cancel {11!}} \times \dfrac {\cancel {(888)!} \times 12 \times \cancel {11!}}{900 \times \cancel {899!}} \\ &=\dfrac {12}{900} \\ &\boxed {\color{#c34632} \approx \%1.3} \end{align*}

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