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?
$\frac{{ }_7 C_2}{{ }_{12} C_2}$
$\frac{{ }_7 P _2}{{ }_{12} P _2}$
$\frac{\left({ }_3 C_1\right)\left({ }_4 C_1\right)}{{ }_{12} C_2}$
$\frac{\left({ }_3 P_1\right)\left({ }_4 P_1\right)}{12 P_2}$
Answer (1)
Calculate the number of ways to choose one art elective: 3 C 1 = 3 .
Calculate the number of ways to choose one history elective: 4 C 1 = 4 .
Calculate the total number of ways to choose two electives: 12 C 2 = 66 .
The probability is the ratio of favorable outcomes to total outcomes: 12 C 2 ( 3 C 1 ) ( 4 C 1 ) = 12 C 2 ( 3 C 1 ) ( 4 C 1 ) .
Explanation
Understand the problem We need to determine the expression that represents the probability that a student chooses one art elective and one history elective from the available electives.
Analyze the given data There are 3 art electives, 4 history electives, and 5 computer electives. The total number of electives is 3 + 4 + 5 = 12 . Each student chooses two electives. We want to find the probability that a student chooses one art elective and one history elective.
Define the probability The probability is calculated as the number of ways to choose one art elective and one history elective, divided by the total number of ways to choose two electives from all available electives.
Calculate the number of ways to choose one art elective The number of ways to choose one art elective from 3 art electives is given by the combination formula 3 C 1 , which is ( 1 3 ) = 1 ! ( 3 − 1 )! 3 ! = 1 ! 2 ! 3 ! = ( 1 ) ( 2 × 1 ) 3 × 2 × 1 = 3 .
Calculate the number of ways to choose one history elective The number of ways to choose one history elective from 4 history electives is given by the combination formula 4 C 1 , which is ( 1 4 ) = 1 ! ( 4 − 1 )! 4 ! = 1 ! 3 ! 4 ! = ( 1 ) ( 3 × 2 × 1 ) 4 × 3 × 2 × 1 = 4 .
Calculate the total number of ways to choose two electives The number of ways to choose two electives from the total 12 electives is given by the combination formula 12 C 2 , which is ( 2 12 ) = 2 ! ( 12 − 2 )! 12 ! = 2 ! 10 ! 12 ! = ( 2 × 1 ) ( 10 !) 12 × 11 × 10 ! = 2 12 × 11 = 6 × 11 = 66 .
Calculate the number of ways to choose one art and one history elective The number of ways to choose one art elective and one history elective is the product of the number of ways to choose one art elective and the number of ways to choose one history elective, which is 3 C 1 × 4 C 1 = 3 × 4 = 12 .
Calculate the probability The probability of choosing one art elective and one history elective is therefore 12 C 2 ( 3 C 1 ) ( 4 C 1 ) = 66 3 × 4 = 66 12 = 11 2 .
State the final answer The expression that represents the probability that a student chooses an art elective and a history elective is 12 C 2 ( 3 C 1 ) ( 4 C 1 ) .
Examples
This type of probability calculation is used in many real-world scenarios, such as determining the likelihood of selecting a specific combination of items from a larger set. For example, if a company wants to form a committee with specific expertise, they can use combinations to calculate the probability of selecting a committee with the desired mix of skills from a pool of employees. This helps in making informed decisions about resource allocation and team composition.
