a 15 member committee will be selected from 18 teachers, 20 engineers, 25 economist,22 environmentalists,and 26 medical doctors. how many 15 members committee can be formed if the committee is composed of 4 teachers, 5 engineers, 3 economists, 2 medical doctors and 1 environmentalist.
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.
To find the number of 15-member committees that can be formed with specific numbers of teachers, engineers, economists, medical doctors, and environmentalists, we need to use the concept of combinations.
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.
To determine the number of possible 15-member committees that can be formed, we need to consider the number of choices for each category of members.
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.