Combinations - Course choosing Someone asked me this question:

Suppose there are $n$ students. Each student must choose 4 courses, and
  every two students can only have a maximum of one common course. How
  many courses are needed to satisfy this condition?

I tried to convert this into a graph theory problem but failed to figure it out.

I presume that what is wanted is the minimum number of courses needed.
For $n=1\;thru\; 4,$ the pattern is $4,7,9,10$. The pattern for $n=5$ is shown below:
$\boxed{1,2}\;\boxed{1,3}\;\boxed{1,4}\;\boxed{1,5}\;\boxed{2,3}\;\boxed{2,4}\;\boxed{2,5}\;\boxed{3,4}\;\boxed{3,5}\;\boxed{4,5}\;$ and it repeats in cycles of $5$
Let number of courses needed for $(n\;mod\; 5)\;be\; k_i,\; i = 1\; thru\; 4$  
Then minimum number of courses needed for $n$ students = $10\times \left[int\left(\dfrac{n}{5}\right)\right] + k_i$
