A high school planning committee needs to select a subcommittee of 3 people from a pool of… — ACT

A high school planning committee needs to select a subcommittee of $3$ people from a pool of $5$ teachers and $4$ students. How many different subcommittees can be formed consisting of exactly $2$ teachers and $1$ student?

Answer: 40 subcommittees

Answer

The total number of different subcommittees that can be formed is 40.

To find the number of unique subcommittees with 2 teachers and 1 student, calculate the combination for choosing 2 teachers from 5, which equals 10, and 1 student from 4, which equals 4. Multiplying these independent choices together results in 40 possible subcommittees.

Step-by-Step Solution

Calculate the number of ways to select 2 teachers from the pool of 5 teachers.

\binom{5}{2} = \frac{5 \times 4}{2 \times 1} = 10

ways

Selection order does not matter for committee members, so combinations are used.

Calculate the number of ways to select 1 student from the pool of 4 students.

\binom{4}{1} = 4

ways

Choosing 1 item from a set of 4 offers 4 distinct possibilities.

Apply the Fundamental Counting Principle to determine total combinations.

10 \times 4 = 40

subcommittees

Multiply the number of choices for teachers by the number of choices for students.

Key Concept

Combinations and Fundamental Counting Principle

Estimated Time:1m 0s

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