To find the probability that a student chooses one art elective and one history elective, we need to follow a series of logical steps. Let's break it down step by step:
1. Total Number of Electives:
We start by calculating the total number of electives available. The school offers:
- 3 art electives
- 4 history electives
- 5 computer electives
Adding these, we get:
[tex]\[
\text{Total electives} = 3 + 4 + 5 = 12
\][/tex]
2. Total Number of Ways to Choose 2 Electives from 12:
The total number of possible ways to choose 2 electives out of 12 can be calculated using combinations. The combination formula for choosing [tex]\( k \)[/tex] items from [tex]\( n \)[/tex] items is given by:
[tex]\[
\binom{n}{k} = \frac{n!}{k!(n-k)!}
\][/tex]
Here, we need to choose 2 electives from 12, so:
[tex]\[
\binom{12}{2} = \frac{12!}{2!(12-2)!} = \frac{12 \cdot 11}{2 \cdot 1} = 66
\][/tex]
3. Number of Ways to Choose 1 Art Elective and 1 History Elective:
To find the number of ways a student can choose 1 art elective and 1 history elective, we multiply the number of ways to choose 1 art elective by the number of ways to choose 1 history elective:
[tex]\[
\binom{3}{1} \times \binom{4}{1} = 3 \times 4 = 12
\][/tex]
4. Calculating the Probability:
The probability that a student chooses one art elective and one history elective is the ratio of the number of favorable outcomes (choosing 1 art and 1 history) to the total number of possible outcomes (choosing any 2 electives from 12):
[tex]\[
\text{Probability} = \frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}} = \frac{12}{66} = \frac{2}{11} \approx 0.18181818181818182
\][/tex]
5. Identifying the Correct Expression:
The expression that represents this probability should match our calculations:
[tex]\[
\frac{\left.\left({ }_3 C_1 \cdot { }_4 C_1\right)}{{ }_{12} C_2}
\][/tex]
Simplified form:
[tex]\[
\frac{3 \cdot 4}{\binom{12}{2}} = \frac{12}{66}
\][/tex]
Thus, the correct expression for the probability that a student chooses one art elective and one history elective is:
[tex]\[
\frac{\left.\left({ }_3 C_1 \cdot { }_4 C_1\right)}{{ }_{12} C_2}
\][/tex]
