The first player shook hands 69 times
The second 69 times less the shake with the first player = 68 times
The third 69 times less the two with the first and second=67 times
the 68th player shook 69 times less 67 times already counted =1
All the skakes of the 69th player are alredy counted
So the total is?
..and this is not the only way of doing this..