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?

A. [tex]\(\frac{{ }_7 C_2}{{ }_{12} C_2}\)[/tex]

B. [tex]\(\frac{P_2}{12 P_2}\)[/tex]

C. [tex]\(\frac{\left({ }_3 C_1\right) \left({ }_4 C_1\right)}{{ }_{12} C_2}\)[/tex]

D. [tex]\(\frac{\left(3 P_1\right) \left(4 P_1\right)}{12 P_2}\)[/tex]



Answer :

Let's determine the probability that a student chooses one art elective and one history elective out of the total number of choices.

We should first find out the number of ways a student can choose one art elective, then one history elective, and finally, the total ways to choose any two electives out of all available electives.

Step-by-step solution:

1. Total electives available:

Students can choose from:
- 3 art electives
- 4 history electives
- 5 computer electives

Thus, the total number of electives available is [tex]\( 3 + 4 + 5 = 12 \)[/tex].

2. Number of ways to choose 1 art elective:

The number of ways to choose one art elective out of three is represented by the combination formula [tex]\({}_n C_r\)[/tex]:
[tex]\[ { }_3 C_1 \][/tex]

3. Number of ways to choose 1 history elective:

The number of ways to choose one history elective out of four is represented by the combination formula:
[tex]\[ { }_4 C_1 \][/tex]

4. Number of ways to choose any 2 electives out of 12 total electives:

The number of ways to choose 2 electives from 12 is given by the combination formula:
[tex]\[ { }_{12} C_2 \][/tex]

5. Probability calculation:

To find the probability that a student chooses one art elective and one history elective, we need to calculate the ratio of the number of favorable outcomes to the total number of possible outcomes:
[tex]\[ \text{Probability} = \frac{{ }_3 C_1 \times { }_4 C_1}{{ }_{12} C_2} \][/tex]

Thus, the expression representing the probability that a student chooses one art elective and one history elective is:
[tex]\[ \boxed{\frac{\left.\left({ }_3 C_1\right) C_1\right)}{{ }_{12} C_2}} \][/tex]