a club has 30 members including 3lawers, 4 teachers and 5 doctors. In how many ways can a committee of 8 be formed to contain 1 teacher, 2 lawyers and 2 doctors?
the categories can be considered separately, so working with the specialties first, there would be 3C2 * 4C1 * 5C2 = 3*4*10 = 120 ways to pick them.
Then, with 5 slots already filled, that leaves only 25 left to choose the other 3 members, or 25C3 = 2300
So, that makes 120 * 2300 = 276,000 ways to form the committee.
To solve this problem, we need to use the concept of combinations.
First, let's identify the number of available candidates for each position in the committee:
- 1 teacher: There are 4 teachers in the club.
- 2 lawyers: There are 3 lawyers in the club, so we need to choose 2 out of them.
- 2 doctors: There are 5 doctors in the club, so we need to choose 2 out of them.
Now, let's calculate the number of combinations for each position:
1. Number of combinations for selecting 1 teacher: C(4, 1) = 4C1 = 4 (choosing 1 out of 4 teachers).
2. Number of combinations for selecting 2 lawyers: C(3, 2) = 3C2 = 3 (choosing 2 out of 3 lawyers).
3. Number of combinations for selecting 2 doctors: C(5, 2) = 5C2 = 10 (choosing 2 out of 5 doctors).
To find the total number of ways a committee can be formed, we need to multiply these individual combinations together:
Total = (Number of combinations for teacher) * (Number of combinations for lawyers) * (Number of combinations for doctors)
= 4 * 3 * 10
= 120.
Therefore, there are 120 different ways to form a committee of 8 members with 1 teacher, 2 lawyers, and 2 doctors from the given club.