"I'm still a bit confused about the conclusion below. Can anyone provide an explanation or proof?" Thanks a lot. " This is a well-known problem with the following solution. Let tot be the total number of objects and mx be the type that has the maximum number of objects (maximum value of sz ). If the number of objects of type mx is at most the number of all other objects (i. e. szmx≤tot−szmx ), then we can pair all objects (except 1 if the total number of objects is odd). Otherwise, we can create tot−szmx pairs of the form (mx,i) for all i except mx . "

Oh, the prove is so detailed and beautiful. thanks a lot.

I was confused about this . Thanks a lot.