The answer is truly 24, but your reasoning is wrong. 24 it is because each handshake includes two people, therefore total of 120 persons participate (though a lot of this participants are equal to each other) in a handshake, and each person did it 5 times, so there must be 24 different persons.
There are many ways to arrange the graph with 60 edges and having each vertex of degree five. You simply chooses two edges from different subcliques in your graph and delete current edges and cross-connect their ends (replace edges $(a,b),(c,d)$ with $(a,c),(b,d)$ ), what you get is a graph with the same number of edges and vertices and same vertex degrees, but different from your original graph. And you can repeat this step many times to create a lot of different graphs.