Question

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? \\(\frac{_7c_2}{_{12}c_2}\\) \\(\frac{_7p_2}{_{12}p_2}\\) \\(\frac{(_3c_1)(_4c_1)}{_{12}c_2}\\) \\(\frac{(_3p_1)(_4p_1)}{_{12}p_2}\\)

Explanation:

Step1: Understand Total Electives

Total electives: 3 (art) + 4 (history) + 5 (computer) = 12. We choose 2, so total combinations: $_{12}C_2$.

Step2: Favorable Outcomes

Choose 1 art from 3: $_{3}C_1$, 1 history from 4: $_{4}C_1$. Favorable: $(_{3}C_1)(_{4}C_1)$.

Step3: Probability Formula

Probability = favorable / total. So $\frac{(_{3}C_1)(_{4}C_1)}{_{12}C_2}$.

Answer:

$\boldsymbol{\frac{(_{3}C_1)(_{4}C_1)}{_{12}C_2}}$ (the third option)