Question

Over the summer a bunch of math teachers from different schools got together for a BBQ. Not one of the teachers knew each other and greets each other with a hearty handshake. How many handshakes will occur at the party if every one of the 15 math teachers shakes hands with each of the others?
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Answer 1

The first partner in the handshake can be any one of 15 people. 
The second partner can be any one of 14 people. 
So the total number of possible arrangements is (15 x 14) = 210 .

BUT . . . .

If Mr. Smith and Mr. Jones are shaking hands, it doesn't matter which one is the first partner and which one is the second partner ...  It's the same handshake either way.

That number of ' 210 ' up above counted both ways separately.  The real number of handshakes between different people is half of that   210/2 = 105 handshakes.