Finding a Company with Specific Friendship Connections

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To determine if it is possible to find a company where each member has exactly five friends and any two members have exactly two mutual friends, we can analyze the friendship connections using lists:

1. Create a list of company members: A, B, C, D, E, F, G, H, I, J.
2. List the number of friends each member has:
   • A: 5 friends
   • B: 5 friends
   • C: 5 friends
   • D: 5 friends
   • E: 5 friends
   • F: 5 friends
   • G: 5 friends
   • H: 5 friends
   • I: 5 friends
   • J: 5 friends
3. List the mutual friends between any two members:
   • A and B: 2 mutual friends
   • A and C: 2 mutual friends
   • A and D: 2 mutual friends
   • A and E: 2 mutual friends
   • A and F: 2 mutual friends
   • A and G: 2 mutual friends
   • A and H: 2 mutual friends
   • A and I: 2 mutual friends
   • A and J: 2 mutual friends
   • B and C: 2 mutual friends
   • B and D: 2 mutual friends
   • B and E: 2 mutual friends
   • B and F: 2 mutual friends
   • B and G: 2 mutual friends
   • B and H: 2 mutual friends
   • B and I: 2 mutual friends
   • B and J: 2 mutual friends
   • C and D: 2 mutual friends
   • C and E: 2 mutual friends
   • C and F: 2 mutual friends
   • C and G: 2 mutual friends
   • C and H: 2 mutual friends
   • C and I: 2 mutual friends
   • C and J: 2 mutual friends
   • D and E: 2 mutual friends
   • D and F: 2 mutual friends
   • D and G: 2 mutual friends
   • D and H: 2 mutual friends
   • D and I: 2 mutual friends
   • D and J: 2 mutual friends
   • E and F: 2 mutual friends
   • E and G: 2 mutual friends
   • E and H: 2 mutual friends
   • E and I: 2 mutual friends
   • E and J: 2 mutual friends
   • F and G: 2 mutual friends
   • F and H: 2 mutual friends
   • F and I: 2 mutual friends
   • F and J: 2 mutual friends
   • G and H: 2 mutual friends
   • G and I: 2 mutual friends
   • G and J: 2 mutual friends
   • H and I: 2 mutual friends
   • H and J: 2 mutual friends
   • I and J: 2 mutual friends
4. Analyzing the lists, we can see that it is not possible to find a company where each member has exactly five friends and any two members have exactly two mutual friends. This is because if each member has five friends, there would be a total of 50 friendships (10 members 5 friends each). However, the total number of mutual friendships listed above is 45 (10 members 9 possible pairs * 2 mutual friends each). Therefore, it is not possible to satisfy both conditions simultaneously.

In conclusion, it is not possible to find a company where each member has exactly five friends and any two members have exactly two mutual friends.