Question

A manager is putting together a team of employees to tackle a special project. The team will consist of a team leader, an assistant to the leader, and 4 other team members whose positions are interchangeable. If there is a pool of 20 employees from which to choose, each of whom would be suitable for any position on the team, how many ways can the manager put together this team?

Answer 1

The total number of ways the manager can form the team from a pool of 20 employees is 1,158,000 ways, calculated by multiplying the number of choices for each role on the team.

Step-by-step explanation:

The student is asking how many ways the manager can assemble a team from a pool of 20 employees for a project. This is a problem of counting and combinatorics which can be solved using the concepts of permutations and combinations.

First, we select the team leader. This can be done in 20 different ways, since there are 20 potential candidates. Next, out of the remaining 19 employees, we select an assistant, which can be done in 19 ways. For the 4 other interchangeable team members, we need to choose 4 out of the remaining 18 employees which is a combination problem. The number of ways to choose 4 employees out of 18 without considering the order is given by the combination formula C(n, k) = n! / (k!(n-k)!), where n is the total number and k is the number to choose. Therefore:

Choose team leader: 20 ways

Choose assistant: 19 ways

Choose 4 team members: C(18,4) ways

Now we compute:

C(18,4) = 18! / (4!(18-4)!) = (18 × 17 × 16 × 15) / (4 × 3 × 2 × 1) = 3060 ways

So, the total number of ways the manager can form the team is 20 (team leader) × 19 (assistant) × 3060 (other members) = 1,158,000 ways.

Answer 2

Answer: 38760 ways

Step-by-step explanation:

Total number= 20

Required team members= 6

Ways= ²⁰₆C

Ways= 38760 ways