number of committees
= C(18,4) x C(20,5) x C(25,3) x C(22,1) x C(26,2)
= 7.8 x 10^14
Why such huge numbers?
Smaller values would have shown the concept more effectively.
= C(18,4) x C(20,5) x C(25,3) x C(22,1) x C(26,2)
= 7.8 x 10^14
Why such huge numbers?
Smaller values would have shown the concept more effectively.
For the teachers, we need to select 4 out of 18.
C(18, 4) = 18! / (4! * (18 - 4)!) = 18! / (4! * 14!) = (18 * 17 * 16 * 15) / (4 * 3 * 2 * 1) = 3060
For the engineers, we need to select 5 out of 20.
C(20, 5) = 20! / (5! * (20 - 5)!) = 20! / (5! * 15!) = (20 * 19 * 18 * 17 * 16) / (5 * 4 * 3 * 2 * 1) = 15,504
For the economists, we need to select 3 out of 25.
C(25, 3) = 25! / (3! * (25 - 3)!) = 25! / (3! * 22!) = (25 * 24 * 23) / (3 * 2 * 1) = 2,300
For the medical doctors, we need to select 2 out of 26.
C(26, 2) = 26! / (2! * (26 - 2)!) = 26! / (2! * 24!) = (26 * 25) / (2 * 1) = 325
For the environmentalist, we need to select 1 out of 22.
C(22, 1) = 22
Finally, to find the total number of 15-member committees, we need to multiply the combinations for each category:
Total = 3060 * 15,504 * 2,300 * 325 * 22 = 79,561,680,000
Therefore, there are 79,561,680,000 different 15-member committees that can be formed from the given constraints.
Given that the committee must consist of 4 teachers, 5 engineers, 3 economists, 2 medical doctors, and 1 environmentalist, we can calculate the number of possible combinations as follows:
1. Selecting the teachers:
The number of ways to choose 4 teachers from 18 is calculated using the combination formula: C(18, 4) = 18! / (4! * (18-4)!) = 3060.
2. Selecting the engineers:
The number of ways to choose 5 engineers from 20 is calculated using the combination formula: C(20, 5) = 20! / (5! * (20-5)!) = 15,504.
3. Selecting the economists:
The number of ways to choose 3 economists from 25 is calculated using the combination formula: C(25, 3) = 25! / (3! * (25-3)!) = 2,300.
4. Selecting the medical doctors:
The number of ways to choose 2 medical doctors from 26 is calculated using the combination formula: C(26, 2) = 26! / (2! * (26-2)!) = 325.
5. Selecting the environmentalist:
The number of ways to choose 1 environmentalist from 22 is calculated using the combination formula: C(22, 1) = 22! / (1! * (22-1)!) = 22.
Now, we multiply the numbers from each category together to find the total number of committees that can be formed:
Total number of committees = 3060 * 15,504 * 2,300 * 325 * 22 = 3,764,320,800,000.
Therefore, there are 3,764,320,800,000 possible 15-member committees that can be formed with 4 teachers, 5 engineers, 3 economists, 2 medical doctors, and 1 environmentalist.