Question 727884
In a group of {{{n}}} people, if every person shakes hands with every other person exactly once, each person would report {{{(n-1)}}} handshakes, and you would get a total of {{{n(n-1)}}} such reports. That number would be double the number of handshakes, because it handshake would be reported twice, with one report by each of the two handshakers involved.
 
If there were 36 handshakes between the {{{n}}} ladies,
{{{n(n-1)/2=36}}} --> {{{n(n-1)=36*2}}} --> {{{n(n-1)=9*4*2}}} --> {{{n(n-1)=9*8}}} --> {{{n=9}}}. There are {{{highlight(9)}}} ladies.

If there were 28 handshakes between the {{{m}}} gentlemen,
{{{m(m-1)/2=28}}} --> {{{m(m-1)=28*2}}} --> {{{m(m-1)=7*4*2}}} --> {{{m(m-1)=7*8}}} --> {{{m=8}}}. There are {{{highlight(8)}}} gentlemen.
 
Each of the 9 ladies shakes hands with each of the 8 gentleman for a total of
{{{9*8=highlight(72)}}} handshakes between a lady and a gentleman.