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A man math xmlns=http://www.w3.org/1998/Math/MathMLmi mathvariant=normalX/mi/math has math xmlns=http://www.w3.org/1998/Math/MathML class=wrs_chemistrymn7/mn/math friends, math xmlns=http://www.w3.org/1998/Math/MathMLmn4/mn/math of them are ladies and math xmlns=http://www.w3.org/1998/Math/MathMLmn3/mn/math are men. His wife \( \mathrm{Y} \) also has math xmlns=http://www.w3.org/1998/Math/MathMLmn7/mn/math friends, math xmlns=http://www.w3.org/1998/Math/MathMLmn3/mn/math of them are ladies and math xmlns=http://www.w3.org/1998/Math/MathMLmn4/mn/math are men. Assume \( \mathrm{X} \) and \( \mathrm{Y} \) have no common friends. The total number of ways in which \( \mathrm{X} \) and \( \mathrm{Y} \) together can throw a party inviting math xmlns=http://www.w3.org/1998/Math/MathMLmn3/mn/math ladies and math xmlns=http://www.w3.org/1998/Math/MathMLmn3/mn/math men, so that math xmlns=http://www.w3.org/1998/Math/MathMLmn3/mn/math friends of each of \( X \) and \( Y \) are in this party, is
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