Editing 216: Romantic Drama Equation

Each straight pair needs to include one of the x males and one of the (n-x) females, so there are x(n-x) possible ways of combining one of each.  E.g., if there are n=5 people, of whom x=2 are male, then there will be 3 possible pairings involving the first male, and 3 possible pairings involving the second, yielding 2(5-2)=6 possible pairs.

Each straight pair needs to include one of the x males and one of the (n-x) females, so there are x(n-x) possible ways of combining one of each.  E.g., if there are n=5 people, of whom x=2 are male, then there will be 3 possible pairings involving the first male, and 3 possible pairings involving the second, yielding 2(5-2)=6 possible pairs.

Each gay pair needs to include either two males or two females.  To choose two males, we can start with any of the x males and choose any of the (x-1) remaining males.  However, that counts each possible pair twice.  E.g., Adam&Steve got counted when we chose Adam first and Steve second, and again when we chose Steve first and Adam second.  To avoid double-counting the pairs, we therefore need to divide the product by 2.  So there are x(x-1)/2 possible pairs of two males.  Similarly, there are (n-x)(n-x-1)/2 possible pairings of two females. Summing these, we get the total number of possible gay pairs as [x^2 - x  +  n^2 - nx - n - xn + x^2 + x]/2.  That simplifies to [n^2 - n  +  2 x^2 - 2 xn]/2.  The left two terms can be combined together as n(n-1) and the right two terms can be combined together as -2x(n-x) or 2x(x-n) [which is negative, because x-n<0].  Since the sum of these terms was divided by 2, we get that the total number of possible gay pairs is n(n-1)/2 - x(n-x), or n(n-1)/2 + x(x-n), which is what the cartoon says.

Each gay pair needs to include either two males or two females.  To choose two males, we can start with any of the x males and choose any of the (x-1) remaining males.  However, that counts each possible pair twice.  E.g., Adam&Steve got counted when we chose Adam first and Steve second, and again when we chose Steve first and Adam second.  To avoid double-counting the pairs, we therefore need to divide the product by 2.  So there are x(x-1)/2 possible pairs of two males.  Similarly, there are (n-x)(n-x-1)/2 possible pairings of two females. Summing these, we get the total number of possible gay pairs as [x^2 - x  +  n^2 - nx - n - xn + x^2 + x]/2.  That simplifies to [n^2 - n  +  2 x^2 - 2 xn]/2.  The left two terms can be combined together as n(n-1) and the right two terms can be combined together as -2x(n-x) or 2x(x-n) [which is negative, because x-n<0].  Since the sum of these terms was divided by 2, we get that the total number of possible gay pairs is n(n-1)/2 - x(n-x), or n(n-1)/2 + x(x-n), which is what the cartoon says.

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