How many handshakes in total?

First of all there is no Probability here, perhaps Combinatorics/Graph theory is more appropriate.

$6$ is also $\binom{4}{2}$ but more importantly it is $3+2+1$.

Thus in all cases the number of hand shakes is $(n-1)+(n-2)+ \cdots + 1 = n(n-1)/2$. Count person by person, ignoring handshakes with people already counted, to see this.

190 is the answer.