How many different committees can be formed from 9 teachers and 39 students if the committee consists of 2 teachers and 3 students? In how many ways can the committee of 5 members be selected?
number of ways = C(9,2) x C(39,3)
= 36(9139)
= ....
number of committees without restrictions
= C(48,5) = 1712304
To find the number of different committees that can be formed, we need to use the combination formula, which is:
nCr = n! / (r! * (n-r)!)
Where "n" represents the total number of people available to choose from, and "r" represents the number of people that need to be selected.
In this case, we have 9 teachers to choose from for the teacher positions, and 39 students to choose from for the student positions. We need to select 2 teachers and 3 students for each committee.
The number of different committees that can be formed can be calculated as follows:
Number of teacher combinations = 9C2 = 9! / (2! * (9-2)!) = 9! / (2! * 7!) = (9 * 8) / (2 * 1) = 36
Number of student combinations = 39C3 = 39! / (3! * (39-3)!) = 39! / (3! * 36!) = (39 * 38 * 37) / (3 * 2 * 1) = 9,139
To find the total number of committees, we need to multiply the number of teacher combinations by the number of student combinations:
Total number of committees = Number of teacher combinations * Number of student combinations = 36 * 9,139 = 328,404
Therefore, there are 328,404 different committees that can be formed from 9 teachers and 39 students if the committee consists of 2 teachers and 3 students.
For the second question, the committee of 5 members can be selected in the same way using the combination formula:
Total number of combinations = (9 + 39)C5 = (48)C5 = 48! / (5! * (48-5)!)
By calculating this, we would find the total number of ways the committee of 5 members can be selected.
To find the number of different committees that can be formed from 9 teachers and 39 students, where each committee consists of 2 teachers and 3 students, we need to use the concept of combinations.
The number of ways to select a 2-teacher committee from 9 teachers can be calculated using the combination formula, which is:
C(n, r) = n! / (r! * (n-r)!)
Where n is the total number of items to choose from, and r is the number of items being chosen.
So, the number of ways to select a 2-teacher committee from 9 teachers is:
C(9, 2) = 9! / (2! * (9-2)!)
= (9 * 8) / (2 * 1)
= 36
Similarly, the number of ways to select a 3-student committee from 39 students is:
C(39, 3) = 39! / (3! * (39-3)!)
= (39 * 38 * 37) / (3 * 2 * 1)
= 9139
To find the total number of ways to form a committee of 2 teachers and 3 students, we need to multiply both counts together:
Total number of ways = Number of ways to select 2 teachers * Number of ways to select 3 students
= 36 * 9139
= 329,304
Therefore, there are 329,304 different committees that can be formed from 9 teachers and 39 students, where each committee consists of 2 teachers and 3 students.
Alternatively, you can also calculate the total number of ways to form a committee of 5 members by directly using the combination formula:
C(48, 5) = 48! / (5! * (48-5)!)
= (48 * 47 * 46 * 45 * 44) / (5 * 4 * 3 * 2 * 1)
= 1,712,304
So, there are 1,712,304 ways to form a committee of 5 members from a group of 9 teachers and 39 students.