Answer these questions based on the following information.
A team of 8 software engineers work on a project, for an American client, in Bengaluru, India. Every team member wants to go to the client's site in USA, as many times as it's possible for them. However, as per the company's policy every member will visit the site same number of times in a year. Each team member has to travel in groups of the same size each time the team goes to client's site. Not all members can be same in any two groups visiting the client's site in a year.
Maximum how many times a team member can go to the client's site in a year ?

Options

A. A) 32
B. B) 35
C. C) 49
D. D) 7

Answer

To solve the problem, we need to determine the maximum number of times a team member can visit the client's site in a year, given the constraints of the team structure and travel policy.

### Step-by-Step Solution:

1. **Identify the Total Team Members**:
The total number of software engineers in the team is 8.

2. **Understand Group Size Requirement**:
Each team member must travel in groups of the same size each time they visit the client's site.

3. **Determine Maximum Group Size**:
Since all team members must visit the site the same number of times, and not all members can be the same in any two groups, we can maximize the group size. The best way to do this is to form groups of half the total members when the total is even. Here, \( n = 8 \) (which is even), so the maximum group size \( m \) is:
\[ m = \frac{n}{2} = \frac{8}{2} = 4 \]
This means each group will consist of 4 members.

4. **Calculate the Number of Unique Groups**:
To find out how many unique groups of 4 can be formed from 8 members, we use the combination formula \( nCr \):
\[ 8C4 = \frac{8!}{4!(8-4)!} = \frac{8!}{4!4!} \]
Calculating this gives:
\[ 8C4 = \frac{8 \times 7 \times 6 \times 5}{4 \times 3 \times 2 \times 1} = \frac{1680}{24} = 70 \]
So, there are 70 unique groups of 4 members.

5. **Determine Maximum Visits per Member**:
Each member can be part of a group of 4. If we focus on one specific member, we need to find out how many different groups that member can be part of. If one member is fixed in a group, we need to choose 3 more members from the remaining 7 members:
\[ 7C3 = \frac{7!}{3!(7-3)!} = \frac{7!}{3!4!} \]
Calculating this gives:
\[ 7C3 = \frac{7 \times 6 \times 5}{3 \times 2 \times 1} = \frac{210}{6} = 35 \]
Therefore, each team member can visit the client's site a maximum of 35 times in a year.

### Final Answer:
The maximum number of times a team member can go to the client's site in a year is **35**.

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