6 teachers and 12 students volunteer for a committee to discuss extra curricular activities. How many committees of 5 people can be made if there must be 3 students on the committee

Answer 1

To form a committee of 5 people with 3 students, we need to choose 3 students from the 12 available and 2 teachers from the 6 available.

The number of ways to choose 3 students from 12 is calculated as 12 choose 3, which can be written as 12C3 or denoted mathematically as ${12 \choose 3}$.

Using the combination formula,

${12 \choose 3} = \frac{12!}{3!(12-3)!} = \frac{12!}{3!9!} = \frac{12\times11\times10}{3\times2\times1} = 220$.

To choose 2 teachers from 6, we can use the same formula as above:

${6 \choose 2} = \frac{6!}{2!(6-2)!} = \frac{6!}{2!4!} = \frac{6\times5}{2\times1} = 15$.

To calculate the total number of possible committees of 5 people with 3 students, we multiply the number of ways to choose the students by the number of ways to choose the teachers:

$220 \times 15 = 3300$.

Therefore, there can be 3300 committees of 5 people that include 3 students.