Question 3797
 This is a basic fact of graph theory.
 
 Every person (not every man !) as a vertex and a handshakes between two
 peresons mean the two vertices connecting by an edge in the graph. In
 other word, for each edge (handshake) corresponding to two person.)

  For each vertex v(person), the number of vertices adjacent to it is
  called the degree (# of his/her handshakes). 
  For any graph G,(ieforany group of people), the summation of deg(v)
  for all vertex in G must be equal to 2|E| (where |E| is the number
  of edges in G, ie #  of handshakes.)  
 
  Next among all persons (all vertices in the graph G), some has
  odd deg while others are of even degree. But, the # of vertices
  with odd degree (odd # of handshakes) should be even. For otherwise,
  the total degree of the odd vertices would be odd (why ?). Thus,
  adding the even vertices would cause the total degree becoming odd.
  This contradicts to the previous claim. 
  Therefore, the number of people who have made an odd number of handshakes is 
  even.
 
  Further questions are welcome.

  Kenny