Smallest number to appear n times in Pascal's triangle

[Marc LeBrun]

 > Do you know if the number of times a binomial coefficient greater than 1 
appears is bounded?

I may be mistaken, but I think so.  choose(n,k) grows like O(n^k), 
right?  So they will eventually exceed any given C.  The first to go will 
be the central column, the last to sail over the horizon will be the pair 
at C=choose(C,1)=choose(C,C-1).  So there's a bounded region which contains 
all instances of C.

However this bound doesn't tell us exactly how many copies of C actually 
occur in that corner of Pascal's triangle.  It also tells us nothing about 
whether or not *some* C occurs exactly five (or however many) times.

Interesting problem...