Prove the following:

(*)       (2n choose n+1)  =  \sum_{k=0}^{n-1}  (n choose k)(n choose k+1)




The student will have to use a combinatorial proof here. Note that
this is similar to the familiar 
      
(+)       (2n choose n)    = \sum_{k=0}^n  (n choose k)(n choose k)

which some students may have learn how to prove, but it is not exactly
the same. A slight modification of the argument for (+) is enough to
prove (*), and can be done only by a student that understands what is
going on, I think.