At a high school, students can choose between three art electives, four history electives, and five computer electives. Each student can choose two electives. Which expression represents the probability that a student chooses an art elective and a history elective?
Answer
A
the fraction with numerator sub 7 cap c sub 2 and denominator sub 12 cap c sub 2
B
the fraction with numerator open paren sub 3 cap p sub 1 close paren times open paren sub 4 cap p sub 1 close paren and denominator sub 12 cap p sub 2
C
the fraction with numerator open paren sub 3 cap c sub 1 close paren times open paren sub 4 cap c sub 1 close paren and denominator sub 12 cap c sub 2
D
the fraction with numerator sub 7 cap p sub 2 and denominator sub 12 cap p sub 2
Explanation
There are 12 electives altogether, and choosing 2 electives without regard to order gives \(\binom{12}{2}\) possible pairs. To get one art and one history elective, choose 1 of the 3 art electives and 1 of the 4 history electives, giving \(\binom{3}{1}\binom{4}{1}\) favorable pairs. Thus the probability is the fraction shown in choice A.
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