A committee of 2 teachers and 3 students is to be randomly chosen from a group of 5 teacher and 6 students. the total number of different committees is?
Ways to choose 2 teachers from 5:
5!/[2!(3!)] = 5*4/2 = 10
ways to choose 3 students from 6
6!/[3!(3!)] = 6*5*4 /(3*2) = 20
10*20 = 200
2 committees with one teacher left over
To find the total number of different committees that can be formed, we need to calculate the combination of selecting 2 teachers out of 5 and 3 students out of 6.
The number of ways to choose 2 teachers out of 5 is denoted by C(5,2) or as 5C2 and can be calculated as:
C(5,2) = 5! / (2! * (5-2)!) = (5 * 4) / (2 * 1) = 10
Similarly, the number of ways to choose 3 students out of 6 is denoted by C(6,3) or as 6C3 and can be calculated as:
C(6,3) = 6! / (3! * (6-3)!) = (6 * 5 * 4) / (3 * 2 * 1) = 20
To find the total number of different committees, we multiply these two combinations together:
Total number of different committees = C(5,2) * C(6,3) = 10 * 20 = 200
Therefore, the total number of different committees that can be formed is 200.
To find the total number of different committees that can be chosen, we can use the concept of combinations.
To choose a committee of 2 teachers and 3 students from a group of 5 teachers and 6 students, we first need to calculate the number of combinations for each category separately, and then multiply them together to find the total number of different committees.
The number of combinations for choosing 2 teachers out of 5 can be calculated using the formula for combinations:
C(n, r) = n! / (r! * (n-r)!)
where n is the total number of teachers (5) and r is the number of teachers we want to choose (2).
So, for the teachers, we have:
C(5, 2) = 5! / (2! * (5-2)!)
= 5! / (2! * 3!)
= (5 * 4 * 3!) / (2! * 3!)
= (5 * 4) / 2!
= 10
Therefore, there are 10 different combinations of choosing 2 teachers from 5.
Similarly, for the students, we have:
C(6, 3) = 6! / (3! * (6-3)!)
= 6! / (3! * 3!)
= (6 * 5 * 4 * 3!) / (3! * 3!)
= (6 * 5 * 4) / 3!
= 20
Therefore, there are 20 different combinations of choosing 3 students from 6.
To find the total number of different committees, we multiply the number of teacher combinations by the number of student combinations:
Total number of different committees = 10 * 20
= 200
Hence, there are 200 different committees that can be chosen from a group of 5 teachers and 6 students, where each committee consists of 2 teachers and 3 students.