Four people in a room each shakes hands with each of the others once how many handshakes are there

The first partner can be any one of the 4 people in the room.
             For each of those . . .
The second partner can be any one of the remaining 3 .

So, the number of possible ways to put together a handshake is  (4 x 3) = 12 .

BUT . . .

It makes no difference whether you put 'A' together with 'B'
or 'B' together with 'A', it's the same handshake both ways.

So although there are 12 ways to build a handshake,
there are only (12/2) = 6 different pairs of partners.