Find the best answers to your questions with the help of IDNLearn.com's expert contributors. Join our community to receive prompt and reliable responses to your questions from experienced professionals.

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]


Sagot :

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]
Thank you for using this platform to share and learn. Don't hesitate to keep asking and answering. We value every contribution you make. Your search for solutions ends at IDNLearn.com. Thank you for visiting, and we look forward to helping you again.