At a party, each boy was supposed to shake hands with each of the other boys present exactly once. Each girl present was supposed to shake hands once with each of the other girls present. No members of the opposite sex were sopposed to shake hands. We had more boys than girls and altogether we had 27 handshakes. How many boys were at the party?

is it 7 boys?

suppose there are x boys ans y girls

no. of handshakes between the boys= ^xC₂= x!/2!.(x-2)! = x(x-1)/2

similarly no. of handshakes between girls = y(y-1)/2

according 2 the ques.

x(x-1)/2 + y(y-1)/2 = 27

x(x-1) + y(y-1)=54

here the only possible positive integer solution is 7 and 4 i.e if x is 7 y is 4 and inverse

since in the ques. no of boys is given more so there are 7 boys in the party