#### Answer

10 handshakes

#### Work Step by Step

Let's look at the handshakes of each person, one by one, without mentioning the same handshake twice.
Person 1 shakes hands with person 2, 3, 4, and 5.
Person 2 also shakes hands with person 3, 4, 5.
Person 3 also shakes hands with person 4, 5.
Person 4 and person 5 shake hands.
The total number of exchanged handshakes is $4+3+2+1=10$.