Imagine a group of 105 women and 95 men randomly divided into two rows, 100 persons each, facing each other.
Let all the members of one row shake hands with the corresponding member of the other row.
Denote the number of woman-to-woman handshakes by W and the number of man-to-man handshakes by M.
Show that the value of W-M is independent of the genders' distribution within the rows.
How much is it ?
(In reply to solution (spoiler)
For every two men who shake hands with each other, there will be two women who then necessarily are freed to shake hands with each other. Both M and W each go up by one and the difference remains the same: 5.