• 59. The table indicates the grade (10 or 11) and high school (North or South) of the 270 students enrolled in Algebra II in the Green City School District.
Suppose 2 of these students will be chosen at random to represent the Algebra II classes at a local STEM (Science, Technology, Engineering, and Mathematics) event. Which of the following expressions gives the probability that both chosen students will be from the same grade and the same high school?
    Choice A

    The number of ways the 2 chosen students are from Grade 10, North \(= {}_{47} C_{2}=\frac{47!}{2!}\)

    The number of ways the 2 chosen students are from Grade 10, South = \({}_{93} C_{2}=\frac{93!}{2!}\)

    The number of ways the 2 chosen students are from Grade 11, North = \({}_{73} C_{2}=\frac{73!}{2!}\)

    The number of ways the 2 chosen students are from Grade 11, South = \({}_{57} C_{2}=\frac{57!}{2!}\)

    The total number of ways 2 students are chosen = \({}_{270} C_{2}=\frac{270!}{2!}\)

    $$\therefore P(same\ grade\ and\ same\ school) \approx \frac{\frac{47!}{2!}+\frac{93!}{2!}+\frac{73!}{2!}+\frac{57!}{2!}}{\frac{270!}{2!}}$$